Reasoning about Shapes and Measurement
Test-Bank Items
Below we have listed assessment items for most sections of Reasoning about Shapes and
Measurement. The items were selected from those that instructors used while piloting the
materials. The number of items is related both to the number of times that the particular
section was piloted and to the emphasis given to the material. For some sections, there are
very few items. Space here, of course, is reduced from that provided on actual tests or
quizzes. Answers are given after all of the items.
Please note: We often use the following directions for true/false items on exams–
For each of the statements below indicate whether the statement is True or False by
CIRCLING the proper word. IF THE STATEMENT IS FALSE, THEN BRIEFLY
EXPLAIN WHY IT IS FALSE OR RESTATE IT SO THAT IT IS TRUE.
16.1-12.2 Shoeboxes Have Faces and Nets; Introduction to Polyhedra
1. T
F Every n-gonal pyramid has n + 1 faces.
1'. T
F Every n-gonal pyramid has n + 1 vertices.
2. T F It is impossible for the number of edges, the number of faces, and the
number of vertices to all be odd numbers, no matter what polyhedron you have.
(Compare #7.)
3. Complete the following:
a. A prism with 16 vertices is (what kind of?) __________________ prism.
b. A hexagonal prism has ______ edges, ______ vertices, and ______ faces.
c. A pyramid with 60 edges has ______ vertices and ______ faces.
d. A pentagonal prism has ______ edges, ______ vertices, and _____faces.
4. Complete the following:
a. A hexagonal prism has _______ lateral edges, _______ edges in all, and
________ vertices.
b. A rectangular pyramid has ________ edges, _______ vertices, and ________
faces.
c. A prism with 9 faces has ________ vertices and _________ edges.
Shapes and Measurement
Test-Bank Items
page 1
d. A pyramid with 32 edges has _________ faces and _________ vertices.
e. A polyhedron with 6 vertices and 8 faces has ________ edges.
5. I am a polyhedron with a total of 9 faces. My 8 lateral faces are isosceles triangles.
What is my full name? ________ ______________ ____________
6. How many faces, edges, and vertices does a 100-gonal prism have? Justify two of
the three counts, without using a formula.
Number of faces _____________
Justification:
Number of edges ______________
Justification:
Number of vertices _______________
Justification:
7. Is it possible for the number of edges, the number of faces, and the number of
vertices of a polyhedron to all be even? If so, give an example. If not, explain why
not.
8. Fill in the blanks:
a. A pyramid with 16 edges has ________ faces and ________ vertices.
b. A polyhedron with 14 faces and 21 edges has ________ vertices.
c. A polyhedron with 20 faces and 12 vertices has ________ edges.
d. A polyhedron with 11 vertices and 20 edges has ________ faces.
9. A hexagonal pyramid has how many edges?
A. 5
B. 6
C. 10
D. 12
E. None of A-D
10. A polyhedron with 14 vertices and 6 faces will have how many edges?
A. 5
B. 6
C. 8
D. 18
E. Not enough information
11. The sum of all the angles in a dodecagon is ____ degrees.
A. 360
B. 1800 C. 150
D. 2160
12. Name the polyhedron that the net below gives.
Shapes and Measurement
Test-Bank Items
page 2
A. heptagon
B. rectangular prism
C. rectangular pyramid
D. triangular prism
E. none of above
Instructor: Other possibilities for #12 (will need to change choices) or leave as a
non-multiple-choice item--
13. I am a polyhedron with a total of 9 faces. My 8 lateral faces are isosceles
triangles, all alike. My best name is:
A. 9-gon
B. octagonal pyramid
D. octagonal prism E. regular octagonal pyramid
pyramid
C. 9-gonal pyramid
F. triangular
14. Fill in the blanks:
a. A pentagonal prism has ______ vertices.
Shapes and Measurement
Test-Bank Items
page 3
b. A 70-gonal pyramid has ______ edges.
c. An n-gonal prism has ______lateral edges.
d. A polyhedron that has 18 faces and 32 vertices has ________edges.
e. The lateral faces of pyramids are what shape(s)? __________________
15. Complete the following with the correct numbers.
a. An octagonal prism has ____edges (in all) and _____ vertices.
b. An n-gonal pyramid has _____ lateral edges and ______ faces.
c. A prism that has 12 faces in all has _____ vertices.
d. A polyhedron with 250 edges and 102 vertices has ______ faces.
e. The sum of the measures of all angles in any 10-gon is _____ degrees.
16. State Euler's formula for polyhedra.
17. What is the sum of the number of edges and the number of vertices of an
octagonal pyramid?
A. 25
B. 22
C. 9
D. 23
E. None of A-D
18. Which of these is safe to say about every square pyramid?
X. Every lateral edge has the same length.
Y. All of the angles on each face are equal.
Z. Every base edge has the same length.
A. X only
B. Y only
C. Z only
D. X and Z only
E. Some choice among X, Y, and Z not given here.
(Compare #22, #24.)
19. What is the best name for the shape below?
Shapes and Measurement
Test-Bank Items
page 4
A. polyhedron
B. eight-gon
C. hexagonal prism
D. octahedron
E. L-pyramid
20. How many faces, edges, and vertices does a 100-gonal prism have?
i. The number of faces is...
A. 101
B. 102
C. 300
D. 200
E. none of the above
ii. The number of edges is...
A. 101
B. 102
C. 300
D. 200
E. none of the above
D. 200
E. none of the above
iii. The number of vertices is...
A. 101
B. 102
C. 300
21. A rectangular pyramid has how many edges, faces, and vertices?
i. The number of edges is:
A.
8
B. 12
C. 5
D. 6
ii. The number of faces is:
A.
8
B. 12
C. 5
D. 6
iii. The number of vertices is:
A.
8
B. 12
C. 5
D. 6
22. Which of these is true for every square prism?
X. Every angle in every face is a right angle.
Y. Every lateral edge has the same length.
Z. All the edges have the same length.
A. X and Y only
Shapes and Measurement
B. X and Z only
Test-Bank Items
C. Y and Z only
page 5
D. X, Y, and Z
E. None of A-D
(Compare #18, #24.)
23. What is the sum of the number of faces and the number of vertices of an n-gonal
prism?
A. 2n
B. n+2
C. 3n
D. 3n+2
E. None of A-D
24. Which of these is true about every right rectangular prism (rectangular solid)?
X. All the edges have the same length.
Y. All the angles on each face have the same size.
Z. Each edge is perpendicular to the edges it meets.
A. X only
B. Y only
C. Z only
D. Y and Z only
E. none of A-D
25. Of the following, which is the best technical name for a filing cabinet?
A. box
B. square box
C. prism
D. right prism
E. square prism
26. Check each one that is a net for a cube.
A. _____
B. _____
C. _____
D. _____
27. Sketch a net for a right rectangular prism. (If your sketch is "off" in important
ways, write your intentions as well.)
16.3 Representing and Visualizing Polyhedra
1. Give the best name for the 3-D shapes represented by the following pictures.
Angles that look like right angles ARE right angles in the 3D shape.
Shapes and Measurement
Test-Bank Items
page 6
b.
a.
b.
a.
_____________________
c.
c.
______________________
d.
(a net)
(a net)
_____________________
d.
______________________
e.
f. (a net)
___________________
Shapes and Measurement
_______________________
Test-Bank Items
page 7
g.
h.
__________________
__________________
2. Give the best name for the shape represented by . . .
a.
b.
3. Draw, as carefully as possible, the following figures:
a. a right pentagonal prism
b. a triangular pyramid
4. Draw a net for a cube in two different ways.
5. On the isometric dot paper, sketch a diagram of a shape that has the following
views:
(left) Front:
Shapes and Measurement
Right:
Test-Bank Items
Top:
page 8
6. How many faces cannot be seen in this drawing of a pentagonal pyramid?
6'. How many faces cannot be seen in this drawing of a hexagonal pyramid?
6". Sketch a pentagonal pyramid. Be sure to include the hidden edges. Is your
pyramid right or oblique?
7. Use the following net to make a 3-D drawing of the shape shown. Use hidden
lines where necessary.
8. Sketch the listed views for the 3-dimensional drawing shown below: (Instructor:
Choose one.)
Shapes and Measurement
Test-Bank Items
page 9
front
right
top
9. For Model X on your table… (Instructor: Notice pre-test prep necessary.)
a. draw a net. (You are not allowed to take the model on your table apart but feel
free to pick it up.)
b. draw a picture of the model.
c. fill in the blanks. (Three words)
The correct mathematical name for model X is a/an
_____________ ______________ ______________
d. for Model Y on your table draw a net.
10. a. How many vertices does a 50-gonal pyramid have? Explain how you figured
it out.
b. How many edges does a 50-gonal pyramid have? Explain how you figured it
out.
16.4 Congruent Polyhedra
1. Which of the following are congruent? If none are congruent, explain how you
know.
Shapes and Measurement
Test-Bank Items
page 10
a.
b.
c.
d.
2. a. On the isometric grid paper, sketch a 3D shape with the following views.
Interpret "front" to be "front left," and "right" to be "front right."
front
right
top
b. On the isometric grid paper below, sketch a shape congruent to the one above,
but from a different viewpoint.
16.5 Some Special Polyhedra
1. a. There are ____ types of regular polyhedra.
Shapes and Measurement
Test-Bank Items
page 11
b. Name the types.
2. Explain why a cube is a regular polyhedron.
3. What is a "regular polyhedron"?
4. Circle T for always true, F otherwise. For each F, explain why.
a. Each edge of a regular polyhedron has the same length.
b. One of the types of regular polyhedra was discovered only in 1914.
c. Every cube is a regular polyhedron.
d. Every regular polyhedron is a cube.
e. Close approximations to regular polyhedra do occur in nature.
5. Name 2 platonic solids, name the regular polygon each is made up of, and then
list the numbers of vertices, edges, and faces for each.
Name of Solid
Polygon used for faces
# Vertices
# Edges
# Faces
17.1-17.3 Review of Polygon Vocabulary; Organizing Shapes;
Triangles and Quadrilaterals
1. Draw a diagram showing the hierarchy of the various quadrilaterals we have
looked at in this course. (Hint: Try a tree diagram like the one in the book.)
2. For each of the following, sketch an example if it is possible. If an example is
impossible, say so.
a. An isosceles triangle that is not an acute triangle.
b. A quadrilateral with two 90-degree angles that is not a rectangle. Be sure to
mark the 90-degree angles.
c. A kite that is also a rectangle.
d. A triangular right prism (show every hidden edge as a dashed segment).
3. For each of the following, sketch an example if it is possible. If it is impossible,
say so and explain why.
a. A trapezoid with exactly one right angle.
Shapes and Measurement
Test-Bank Items
page 12
b. A parallelogram that is an isosceles trapezoid.
c. A pentagon with exactly one right angle.
d. An isosceles obtuse triangle.
e. A rhombus that is not a kite.
(Compare #9, #10, #14.)
4. What is the sum of the measures of the exterior angles, one at each vertex, of
every convex polygon? Explain your reasoning for credit. (Instructor: The term
"convex" was in the Exercises only.)
5. Tell whether each statement is always true, sometimes true, or never true. Support
your decisions.
a. The diagonals of a parallelogram bisect each other. ____________________
b. The diagonals of a kite bisect each other. ___________________________
6. Indicate whether each of the following is always true, sometimes true, or never
true.
a.
A parallelogram is an isosceles trapezoid ____________________
b.
A square is a rhombus __________________________________
c.
A scalene triangle is an acute triangle _______________________
7. Consider the following definitions of a trapezoid:
Definition A: A quadrilateral with at least one pair of opposite sides parallel.
Definition B: A quadrilateral with exactly one pair of opposite sides parallel.
Is it possible to draw a figure that is a trapezoid according to Definition A, but is
not a trapezoid according to Definition B?
If yes, draw a figure that satisfies the conditions, and explain why your figure
satisfies those conditions.
If no, explain why not.
8. T F
Every square is a special quadrilateral.
T F Every square is a rectangle.
T F Every rectangle is a parallelogram.
Shapes and Measurement
Test-Bank Items
page 13
T F Every rhombus is a parallelogram.
T F Every rectangle is a special square.
T F Every trapezoid is a special parallelogram.
T F Any fact that is true for every parallelogram is also true for every square.
T F Any fact that is true for every rectangle is also true for every quadrilateral.
9. Draw (if possible) an isosceles trapezoid with EXACTLY one right angle. If it is
not possible, explain why.
10. Draw (if possible) a kite that does not have four congruent sides. If it is not
possible, explain why.
11. Give the best “name” (if the shape is possible) for the descriptions listed below.
If the shape is not possible, explain why.
a. A kite which is also an isosceles trapezoid:
b. A rhombus that is not equilateral:
c. An equilateral (but not regular) isosceles trapezoid:
12. Circle the correct answer—True (T) or False (F).
a. Every kite is also a parallelogram
T
F
b. There are a total of 1175 diagonals in a 50-gon.
T
F
isosceles trapezoid as a base.
T
F
d. A pentagonal pyramid has a total of 5 vertices.
T
F
regions.
T
F
f. A rhombus is a kite.
T
F
T
F
c. It is possible to make a regular pyramid using an
e. The lateral faces of a prism are always parallelogram
g. The interior angles of a trapezoid add up to 360,
but only 270 if it is an isosceles trapezoid.
13. Give the best names for the following shapes.
a. A regular quadrilateral: ______________
Shapes and Measurement
Test-Bank Items
page 14
b. An isosceles trapezoid with at least one right angle:______________
14. If possible, sketch an example of each. If it is not possible, explain why. Use
tick marks, hidden edges and label equal angles to make your intent clear.
a. A trapezoid with exactly 2 right angles.
b. A kite with equal diagonals.
15. For the following conjecture, draw an example that supports the claim and also
one that shows the claim is false.
“The diagonals of a parallelogram are equal”
True Example
Counterexample
16. Which of the following shapes will have all of the properties that every isosceles
trapezoid has?
A. parallelogram
B. kite
C. rhombus
D. rectangle
17. Why is it not safe to make a general conclusion based on a drawing?
18. Why is it not 100% safe to use inductive reasoning?
19. Put the following in the blanks below to show the relationships among the terms:
isosceles trapezoid, parallelogram, quadrilateral, rectangle, rhombus
trapezoid
kite
square
19'. Arrange (only) the following terms in a hierarchical diagram, with the most
general at the top:
Shapes and Measurement
Test-Bank Items
page 15
kite
square
polygon
trapezoid
rectangle
20. Complete the following:
a. The measure of EACH interior angle of a regular decagon is __________ .
b. If an interior angle of a regular polygon is 175 degrees, then the polygon has
_____ sides.
c. The number of congruent sides on a scalene triangle is ________ .
d. The number of diagonals in 16-gon is _____________ .
21. For each of the following, sketch an example if it is possible. (Be sure to mark
your picture to fit.) If it is impossible, say so, and explain why or show a
counterexample.
a. A parallelogram with exactly one right angle.
triangle.
c. A rectangle that is
not a parallelogram.
b. An isosceles right
d. An equilateral quadrilateral that is
not regular.
e. A concave hexagon
22. Classify the following as true or false. If an item is false, sketch a
counterexample.
a. T
F
Every square is a quadrilateral.
b. T
F
Every rectangle is a square.
c. T
F
Every rhombus is a kite.
d. T
F
Every equilateral triangle is an isosceles triangle.
e. T
F
Every trapezoid is a parallelogram.
f. T
F
Any fact that is true for every parallelogram is also true for every
square.
g. T
F
Any fact that is true for every rectangle is also true for every
quadrilateral.
h. T
F
The diagonals of a rhombus are congruent.
i. T
F
The opposite angles of a parallelogram are congruent.
Shapes and Measurement
Test-Bank Items
page 16
j. T
F
The diagonals of a rectangle bisect each other.
23. A student states that a square cannot be a rhombus. What irrelevant
characteristic(s) might she be assuming to be important? How would you help her
to understand her error?
24. How many diagonals does each of these have?
a. a pentagon _______
b. a 103-gon ___________ (Show your work.)
25. The sizes of three interior angles of a quadrilateral are 65˚, 35˚, and 60˚. What is
the size of the fourth angle of the quadrilateral?
A. 20˚
impossible.
B. 100˚
C. 160˚
D. 200˚ E. This quadrilateral is
26. An isosceles triangle has two angles, one with 100˚ and the other with 40˚. How
large is the third angle?
A. 40˚
B. 60˚
C. 100˚
D. 140˚
E. Need more info
27. If a polygon is equiangular, then it must be…
A. equilateral B. regular
C. a triangle D. both A and B
E. none of A-D
28. An angle that is supplementary to an angle with size 70 degrees has what size?
A. 20 degrees B. 70 degrees C. 90 degrees D. 110 degrees
E. 180 degrees
29. Find the number of degrees in each lettered angle.
a
c
120Þ b
a.
b.
70Þ
c.
30. The sum of the sizes of all of the angles of a 17-gon is ___________ degrees.
(Show your work.)
31. For each of the following statements, circle whether the statement is Always True
(AT), Sometimes True (ST) or Never True (NT). If the answer is sometimes true
then draw a picture of a shape that meets both requirements and a shape that
meets only the first requirement.
A Drawing that
Shapes and Measurement
Test-Bank Items
page 17
Meets both
a. A parallelogram is a rectangle. AT ST
NT
b. A rectangle is a parallelogram. AT ST
NT
c. A rhombus is a rectangle.
AT ST
NT
d. A prism is a pyramid.
AT ST
NT
Meets only 1st
32. When asked to find the sum of the interior angles in a hexagon a student writes
the following. Comment on whether the student’s mathematical reasoning is
correct or incorrect. If it is correct, explain how you know. If it is incorrect,
explain what was incorrect about the student's thinking and what he/she would
have to do to correct the error.
33. Sketch an example if it is possible. If any are not, explain why.
a. An equilateral quadrilateral that is not equiangular (What is this shape usually
called?)
b. An equilateral obtuse triangle
c. A pentagon with exactly three right angles.
34. Create a Venn Diagram or a Hierarchy Diagram for only the following terms:
Quadrilaterals, Squares, Rectangles, Polygons, Rhombi
35. (Used as a take-home problem; students had worked with pentominoes in
Exercise 12b. in Section 18.3) How many different arrangements of three identical
rhombi are possible in which each rhombus must match up edge to edge with at
least one other rhombus. Two arrangements are considered the same if one of the
arrangements can be flipped and/or rotated to obtain the other arrangement.
a. Sketch all possible arrangements.
Shapes and Measurement
Test-Bank Items
page 18
b. Explain a counting strategy you used to justify that you have found all the
different arrangements.
36. Why is 10 called a triangular number?
37. Sketch, if possible, an obtuse isosceles triangle that has an angle with 30
degrees. If such a triangle exists, give the sizes of the other angles. If such a
triangle is impossible, explain why.
38. State a fact that is true for all rhombuses, but not true for all kites.
39. State a fact that is true for all rectangles, but is not true for all parallelograms.
40. State two facts that are true for the diagonals of every rhombus.
41. State two facts that are true for the diagonals of every rectangle.
42. State two facts that are true for every parallelogram.
43. Consider the statement: The diagonals of a parallelogram are equal.
Is the statement…
A. always true? B. sometimes true? C. never true?
44.
A square is a rhombus:
A. always
45.
B. sometimes
C. never
The diagonals of every parallelogram....
A. bisect the angles of the parallelogram B. are parallel to each other
C. are perpendicular to each other
D. are equal in length
E. none of A-D
46.
Which of these is true for every rhombus?
I. The diagonals of a rhombus must be equal.
II. The sides of a rhombus must be equal.
A. I only.
47.
B. II only
C. I and II
D. Neither I nor II
If IJKLMN is a regular polygon, then...
A. all the diagonals are equal in length.
Shapes and Measurement
Test-Bank Items
page 19
B. IJKLMN has 7 sides.
C. angle JKL has twice as many degrees as angle LMN.
D. IJKLMN has exactly 7 symmetries.
E.
48.
none of A-D
How are the diagonals of every rectangle related?
X. The diagonals are the same length.
Y. The diagonals are perpendicular.
Z. The diagonals bisect the angles of the rectangle.
A. X only
B.
Y only C.
49.
T
F
Every square is a special quadrilateral.
T
F
Every square is a rectangle.
T
F
Every rectangle is a parallelogram.
T
F
Every rhombus is a parallelogram.
T
F
Every rectangle is a special square.
T
F
Every trapezoid is a special parallelogram.
T
F
Any fact that is true for every parallelogram is also true for every
square.
T
F
Any fact that is true for every rectangle is also true for every
quadrilateral.
50.
Z only D.
X and Z only E.
None of A-D
What is the best name for every four-sided polygon with equal sides?
A. square
B. parallelogram
C. rectangle
D. rhombus
E. kite
51. Fill in the choices on the left side with the BEST choice from the list on the right.
You may re-use a choice.
____ Has parallel lateral edges
____ A regular quadrilateral
____ Has an equal number of faces and vertices
____ A polygon which is never a kite
____ All of its faces could be regular
____ Is always a kite
Shapes and Measurement
Test-Bank Items
a. pyramid
b. non-square rectangle
c. parallelogram
d. square
e. oblique prism
page 20
____ Has lateral faces that are triangular regions
____ Has two bases
52. Is it possible for the sum of the angles of a polygon to be 180,000 degrees?
Explain.
53. What is the measure of one interior angle of a regular 18-gon?
54.
Find the measure of angle A of the triangle.
A
81Þ
B
123 o
C
A.
B.
C.
D.
E.
•
D
55. Which of the following is(are)
IMPOSSIBLE for x, y, and z in the
sketch to the right? (The drawing
is not to scale.)
156˚
24˚
57˚
20˚
not enough information
yÞ
i. x = 100, y = 110, z = 150
xÞ
ii. x = 80, y = 130, z = 150
zÞ
A. Only i is impossible.
B. Only ii is impossible.
C. Both i and ii are impossible.
D. Each of i and ii is possible.
18.1 Symmetry of Shapes in a Plane
1. How many reflection and rotational symmetries does a regular octagon have?
reflection ______________
rotational ________________
2. a. Add to the following design so that it has a reflection symmetry. Draw in the
line of symmetry.
Shapes and Measurement
Test-Bank Items
page 21
b. Now add to the original design so that it has a rotational symmetry. Show the
center of rotation.
(Compare #3, #9, #10.)
3.
Add to the following figure so that it has rotational symmetry. Show a center of
rotation after the addition.
4. a. How many rotational symmetries does a regular 12-gon have?
b. What is the minimum (positive) number of degrees of rotation for a symmetry
of a regular 12-gon?
5. How many reflection symmetries does the figure below have? ___
How many rotational symmetries does it have? ____
6. Fill in the blanks with the correct number.
a. A regular 90-gon has a total of ______rotational symmetries.
b. A regular 90-gon has a total of ______ reflection symmetries.
Shapes and Measurement
Test-Bank Items
page 22
c. The smallest positive rotational symmetry of a regular 90-gon would have
_____ degrees.
7. T F A circle has exactly 360 rotational symmetries.
8. Sketch a regular triangle and mark all lines of reflection symmetry.
9. Add to the following design so that it has a reflection symmetry. Draw in the line
of symmetry.
10. Add to the picture so that it has a rotational symmetry. Show the center of
rotation.
11. How many reflection symmetries does a regular hexagon have? Draw the
hexagon and show all the lines of symmetry.
12. Determine the smallest positive amount of the turn (in degrees) for the rotational
symmetries in each of the figures below:
a.

Shapes and Measurement
b.

Test-Bank Items
c.

page 23
13. For each figure, draw in all lines of reflection symmetry. If the figure has no
lines of symmetry, write “none.”
a.

b.

c.

12-13. Other figures from Word's clip art (no answers provided).
(Ignore the doorknob.)
18.2 Symmetry of Polyhedra
1. Consider the following right regular hexagonal prism:
Shapes and Measurement
Test-Bank Items
page 24
How many reflection symmetries does the figure have? ______
How many rotational symmetries does the figure have? ______
2. a. How many reflection symmetries does every right prism have if its bases are
rectangles that are not squares?
b. How many rotational symmetries does every regular square pyramid have?
3. T F The polyhedron below has exactly one plane of reflection symmetry and
one line of rotational symmetry.
4. Describe all of the rotational and reflection symmetries of the following 3dimensional object. (Feel free to draw pictures to demonstrate.) All of the angles
in the object are 90˚.
19.1 Tessellating the Plane
1. a. Explain how you know that every parallelogram will tessellate the plane. Use
pictures if you want to.
b. Explain how you know that every triangle will tessellate the plane. Use pictures
if you want to.
c. Explain how you know, without taking the time to draw it out accurately, why a
regular pentagon cannot tessellate the plane. (Instructor: Compare #7.)
2. State whether each of the following shapes can tessellate the plane:
a. A regular pentagon yes
no
b. A regular hexagon yes
no
c. This shape
no
Shapes and Measurement
yes
Test-Bank Items
page 25
d. This shape
yes
no
(Compare #4, #5, #6.)
3. Explain why a regular hexagon can tessellate the plane, while a regular octagon
cannot.
4. State whether each of the following shapes can tessellate the plane:
a. A regular 11-gon
yes
no
b.
This shape
yes
no
c.
This shape
yes
no
5. Which of these regions will tessellate the plane?
yes____
no_____
yes_____
no_____
yes_____
no_____
yes_____
no_____
(Regular pentagon)
6. Circle the letter if the shape can tessellate the plane:
a.
A regular octagon
Shapes and Measurement
Test-Bank Items
page 26
b.
A regular triangle
c.
An obtuse scalene triangle
d.
A regular dodecagon
e.
This shape:
7. Explain whether a regular pentagon can tessellate the plane. Be very
specific,using angles measures. Write in full, understandable sentences.
8. Given the following figure, show how to tessellate the plane. Use at least 10
copies, not all in a row.
19.2 Tessellating Space
1. Circle the letter of the shape, if copies of it could tessellate space.
a. A Cheerios cereal box
b. A baseball
c. A carrot (one size)
d. A CD box
20.1-20.2 Size Changes in Planar Figures; More about Similar Figures
Instructor: In making up triangles for quiz/test purposes, one often chooses lengths for
ease of calculation, and then perhaps guesses for other lengths or angle sizes that are
actually impossible. The Auxiliary Folder has an Excel spreadsheet (Triangle Solutions)
that gives correct angle sizes and lengths in SSS, ASA, and SAS situations.
Shapes and Measurement
Test-Bank Items
page 27
1. Are the two shaded triangles below similar? If so, then what is the scale factor? If
not, why are they not similar?
3.48 cm
5.8 cm
4.5 cm
2.7 cm
6.7 cm
4.02 cm
2. Frederick drew a quadrilateral on the chalkboard. He then wanted to draw a new
quadrilateral similar to the original one, with a scale factor of 3. This is what he
did:
3 ft.
1 ft.
2.4 ft.
0.6 ft.
When he was done, Frederick was confused because the new quadrilateral did not
appear to be “three times as large” as the original.
a. Why did Frederick draw the picture the way he did?
b. What (if anything) should Frederick have done differently, and WHY?
3. Equilateral triangle X has sides 8 cm long, and equilateral triangle Y has sides 12
cm long. Are X and Y similar? Explain.
4. Complete the following:
a. Eight feet is ________ times as long as six feet.
Shapes and Measurement
Test-Bank Items
page 28
b. Eight feet is ________ times longer than six feet.
5. Complete the following:
a. 3.8 cm is _________ times longer than 1.1 cm.
b. 24 inches is __________ % as long as 36 inches.
c. A change in the number of graduates from Smalltown High from 20 to 27
represents a ___________ % increase.
6.
T F Every two right triangles are similar.
7.
a. Explain why triangles ACD and AEB below must be similar.
A
x cm
6.1 cm 4 cm
E
36º
7.2 cm
B
5.5 cm
36º
D
b.
98º
y cm
C
Knowing that triangles ACD and AEB are similar, find x and y in the
picture above.
x __________
y __________
8. Circle true or false for each statement. If false, rewrite the statement so that it is
true.
a. True False 6.5 cm is 3.25 times longer than 2 cm.
b. True False A price change from $14 to $21 is an increase of 50%.
c. True False A 75˚ angle is 2.5 times as large as a 30˚ angle.
d. True False 3.5 is about 170% of 1.3.
e. True
False 8 12 inches is 4 14 times longer than 2 inches.
Shapes and Measurement
Test-Bank Items
page 29
f. True
False A change in the cost of a day's stay at a hospital from $200 to
$240 a day is an increase of 120%.
g. True
False A 70˚ angle is 3.5 times as large as a 20˚ angle.
h. True
False If two rectangles have the same area, then they are similar.
i. True
False If two polygons are similar with scale factor 1, then they are
congruent.
9. Triangle ABC is similar to triangle DEC, with E corresponding to B. If you know
that AC = 6 cm in length, AB = 4 cm, CD =10 cm, and CE = 13 cm, find the
lengths of the other segments and the measures of the angles not given. Be sure to
give units.
E
A
85Þ
6 cm
13 cm
4 cm
60Þ
C
B
10 cm
D
length of BC = __________
length of DE = __________
measure of angle BCA=__________
measure of angle ECD=________
measure of angle D=_____________
measure of angle E=___________
10. A 10 cm by 12 cm (width by length) rectangle is the image of a shape having
length 4 cm, for some size change.
a. What is the scale factor of the size transformation?
b. What are the dimensions of the original shape?
c. Are the two quadrilaterals similar? Explain.
_____ because
11. Show another segment…
2 times as long as this one: ___________
Shapes and Measurement
Test-Bank Items
page 30
2 times longer than the given segment in part a.
300% longer than the given segment in part a.
12. Fill in the blank to make each statement a true statement.
a. ______ is 25% of 200.
b. ______ is 200% more than 20.
c. 30 is 1 12 times more than _______.
13. The ruler method was used to enlarge the right triangle below. What is the scale
factor? What is x? Show your work for credit. (Note: This figure is not drawn
to scale.)
Scale factor = ______
3 cm
10 cm
5 cm
x
x = ______
center
14. In map A, the distance from San Diego to Las Vegas is represented as 1.5 inches.
The actual distance is 350 miles. In map B, the distance from San Diego to Las
Vegas is represented as 2 inches and the distance between Las Vegas and Ely,
Nevada, is represented as 3 inches. What is the actual distance between Las
Vegas and Ely? Write enough down to explain your thinking. Hint: A diagram
may help. Don’t forget the units.
15. Suppose triangle ABC is similar to triangle DEF with AB = 6, BC = 5, AC = 7,
and DE = 10, all in centimeters. Assume A  D, B  E, C  F
a. Find the lengths of DF and EF.
b. If the area of triangle ABC = about 15 cm2, what is the area of triangle DEF?
Justify your answer.
16. Fill in the blanks to make each a true statement:
a. Kristin charges 25% less than what Beth charges for tutoring. Beth charges
$30.00, so Kristin charges $______.
b. 12 seconds is 13 as much as ______ seconds.
c. A 48˚ angle is 50% larger than a _____ degree angle.
Shapes and Measurement
Test-Bank Items
page 31
17. Show another segment that is: (Please draw separate segments so the reader can
identify your thinking. Make any marks that would help the reader.)
a. 3 times as long as this one:
Ans:
b. 2 12 times longer than the segment originally given in part a:
Ans:
c. 125% longer than the segment originally given in part a.
Ans:
18. Explain why or why not the following shapes are similar (or are not similar).
a. Every two triangles. _______ because
b. Every two squares. _______ because
19. Triangle A has an area of 50 cm2. Triangle B is similar to Triangle A but has
sides twice as long. What is the area of Triangle B?
20. The ruler method was used to shrink the right triangle below. What is the scale
factor for the length of the two triangles? What is x? Show work to justify your
answer. The figures are not drawn to scale.
Scale factor = ______
x = ______
7 cm
6 cm
5 cm
x
center
10 cm
21. Given the following picture, expand it using a scale factor of 2 and the given
center:
Center

22. The two shapes below are similar. Find the missing angle size, y, and the missing
length, x. Justify your answers.
Shapes and Measurement
Test-Bank Items
page 32
10
90
100
11.25
4
4.5
y
95
12.5
5
75
x
6
23. The triangles are deliberately NOT drawn to correct scale.
6 cm 80Þ 4.5 cm
40Þ
y cm
10 cm
60Þ
x cm
80Þ 4 cm
a. How do you know that the two triangles are similar?
b. Find x and y. (Show your work for credit.)
x = _____
y = _____
24. In a photograph of two buildings, Building P is 5 cm tall and Building Q is 7 cm
tall. In an enlargement of the photograph, Building P is 8 cm tall. How tall is
Building Q in the enlargement?
A. 10 cm
B. 11 15 cm
C. 4 83 cm
D. not enough information
25-26. Use the triangles below to answer the following questions:
Shapes and Measurement
Test-Bank Items
page 33
x
88Þ
62Þ
3.4 cm
4 cm
y
30Þ
4.9 cm
88Þ
2.8 cm
25. What is the length of x, 26. What is the
to the nearest tenth cm?
length of y, to the
nearest tenth cm?
A. 1.9 cm
B. 5.1 cm
A. 6.5 cm
C. 0.3 cm
B. 4.2 cm
D. 1.3 cm
C. 2.2 cm
E. not enough info
D. 5.8 cm
E. not enough
info
27. Refer to the similar figures below. What is the length of x, to the nearest 0.1
cm?

5 cm
9 cm



8 cm
x
100Þ
A.
B.
C.
D.
E.
3.9 cm
12.6 cm
11 cm
12 cm
14.4 cm
100Þ
28.
Two pentagons have the following lengths of sides:
Pentagon A: 7 cm, 7 cm, 7 cm, 7 cm, 10 cm
Pentagon B: 4 cm, 4 cm, 4 cm, 4 cm, 7 cm
Are Pentagons A and B similar? Explain your decision.
29. Triangle A and triangle B are similar. The area of triangle A is 560 square cm.
The area of triangle B is 35 square cm. How long would a segment in triangle A
be, if the corresponding segment in triangle B is 4 cm?
A. 64 cm
B. 16 cm
C. 4 cm
D. 1 cm
E. Not enough info
30. Susan is 166 cm tall and in the afternoon she casts a shadow about 55 cm long.
Her sister, who is standing next to her, casts a shadow about 35 cm long. About
how tall is Susan's sister?
A. 146 cm
Shapes and Measurement
B. 105 cm
C. 95 cm
Test-Bank Items
D. 90 cm
E. 12 cm
page 34
31. A triangle is similar to a larger triangle that has lengths 3 times as long as the
corresponding lengths in the smaller triangle. How large is the angle in the
smaller triangle that corresponds to a 30˚ angle in the larger triangle?
A. 10˚
B. 30˚
C. 90˚
D. 270˚
E. None of A-D
32. What would you have to do to determine whether another quadrilateral EFGH
(not shown) is similar to ABCD below?
C
19 cm
B
110Þ
118Þ
17 cm
63Þ
69Þ
A
20 cm
D
35 cm
33. Treat the measurements in
the drawing to the right as
accurate, even though the
drawings are intentionally
"off."
64Þ
6
cm
116Þ
8 cm
90Þ 90Þ
4 cm
Are the two quadrilaterals
similar? Explain.
12 cm
9 cm
90Þ
90Þ
6.2 cm
_____ because
34. You want to make a larger, similar version of the
trapezoid to the right, so that the 4 cm side will be 10
cm in the larger version.
53Þ
127Þ
7 cm
10 cm
90Þ 90Þ
a. How long will the other three sides of the larger
trapezoid be?
_______ cm
4 cm
_______ cm and _______ cm
b. What are the sizes of the angles in the larger version? ___˚ ___˚ ___˚ and
___˚
20.3 Size Changes in Space Figures
Shapes and Measurement
Test-Bank Items
page 35
1. If the two polyhedra below are similar and oriented alike, then what are the values
of x and y? Show your work for credit.
y inches
3.8 inches
3.4 inches
4.2 inches
x inches
3.6 inches
x = _____________
y = _____________
2. Two regular pentagonal pyramids have the following dimensions:
Length of Base Edge
Length of Lateral Edge
Pyramid A:
7 cm
10 cm
Pyramid B:
4 cm
7 cm
Are the two pyramids (mathematically) similar? _____ Explain your reasoning:
3. A right rectangular prism has a volume of 2 cubic inches. A second right
rectangular prism is similar to the first one and has a volume of 128 cubic inches.
a. What is the scale factor to go from the first prism to the second?
b. What is the scale factor to go from the second prism to the first?
4. If a cube has edges that are 2 cm long, what is the total surface area of a larger
similar cube with a scale factor of 2.3?
5. Suppose you have an L-shaped figure such as the one below. What is the volume
of a figure similar to this one, with a scale factor of 2.8? Write enough to make
your reasoning clear.
Shapes and Measurement
Test-Bank Items
page 36
Alternatives for #5:
6. Cube A has edges that are 3 times as long as the edges of Cube B. The volume of
Cube A is 108 cubic centimeters. What is the volume of Cube B?
7. Each face of the small cubes making the shape below has an area of 1 square unit.
a.
Surface area: ____________
Volume: ____________
b. Suppose another prism is similar to the one above but 2 times larger in all
dimensions. What is the surface area and volume of the new prism? Write
enough to make your thinking clear.
Surface Area:________
Volume:________
Shapes and Measurement
Test-Bank Items
page 37
8. A right rectangular prism with dimensions
enlarged by a scale factor of 1 12 .
2
3
by
4
3
by
8
3
(each in inches) is
a. The volume of the enlarged prism is ___________.
b. The surface area of the original prism is _______ times as much as the surface
area of the larger right rectangular prism.
9. Jeff’s hot tub is 3 feet deep and holds 1500 gallons of water. He’s building a pool
which is similar in shape, but 9 feet deep. How much water will his pool hold?
10. Explain why or why not the following shapes are similar (or not similar).
a. Every two rectangular prisms.
b. Every two cubes.
c. Every two pyramids so long as the base is a square region.
11. Pinocchio's nose grew to 4 times its normal length. If the rest of Pinocchio's body
grew in the same way, what would be his new surface area?
A. 4 times his original surface area
B. 8 times his original surface area
C. 16 times his original surface area D. 64 times his original surface area
E. None of A-D
12. A teacher has a model of an L shaped building made of 4 cubes that snap
together. She wants to make an enlargement, similar in shape to the original. She
has only 600 cubes. What scale factor should she use to make the largest building
possible? Explain your reasoning.
13-14. Consider the shape to the right, made of centimeter cubes
glued together face-to-face.
13. What is the volume of the shape, in cubic centimeters?
A. 11
B. 10
C. 8
D. 6
E. 5
14. A larger shape, similar to the one above, is also made of cubes. The larger shape
is twice as tall, twice as wide, twice and deep, and in general twice as long in
every corresponding length dimension. What is the volume of the larger shape, in
cubic centimeters?
A. 40
Shapes and Measurement
B. 20
C. 16
Test-Bank Items
D. 10
E. None of A-D
page 38
21 Curves, Constructions, and Curved Surfaces
1. Explain how the mathematical ideas of “cylinder” and “cone” are different from
most people’s traditional, everyday use of these terms.
2. Draw, as carefully as possible, the following figures:
a. a right circular cone
b. an oblique elliptical cylinder.
3. Prism is to pyramid as cylinder is to ___________. Explain your response.
4. Tell whether each statement is always true, sometimes true, or never true. Support
your decisions.
a. A chord of a circle may have one endpoint at the center of the circle.
__________
b. A circular cylinder has reflection symmetry. ___________________________
5. How many reflection symmetries does a right circular cylinder have?
6. Circle the correct answer, and explain how you know.
T F
a. Every chord of a circle is a diameter of a circle.
T F
b. The diameter of a circle is the longest chord of a circle.
7. Using a compass and a straightedge construct an isosceles trapezoid that is not
equiangular. Do not erase any important construction marks.
8. Circle the correct answer--T if always true, F otherwise.
a. Every two spheres are similar.
T
F
b. If Earth were a “perfect” sphere, then the equator would be
T
F
c. A diameter is a chord.
T
F
d. A chord is a diameter.
T
F
considered to be a minor circle.
L
M
9. Given the circle shown, name the following:
O
Shapes and Measurement
Test-Bank Items
Ppage 39
N
a minor arc ______
a major arc ______
a diameter ________
Draw a chord and label it AB
Shade in a sector
10. Is this statement always true, sometimes true, or never true: A diameter is a
chord of the circle. Explain.
11. a. Copy segment GR using a compass and a straightedge. Call it JM. Show all
construction marks.
b. Construct the perpendicular bisector of line segment GR. Show all
construction marks.
G
R
12. a. Copy angle O using a compass and a straightedge. Show all construction
marks.
b. Construct the angle bisector of angle O. Show all construction marks.
O
13. With compass and straightedge, construct a rectangle having sides with these two
lengths: ______________ and ______________________ . Do not erase
any key construction marks.
Shapes and Measurement
Test-Bank Items
page 40
14. With compass and straightedge, construct an angle with size 135 degrees. (Hint:
135 = 90 +…..)
15. With compass and straightedge (and without protractor), construct an angle with
size 1.5 times the size of angle O. Do not erase any key construction marks.
O
16. With compass and straightedge, construct a line through point P that is parallel to
line m. Do not erase any key construction marks.
P
m
17. Give the sizes of the lettered angles in the following figure. Do not forget to give
the units.
x=
y
z
x
100Þ
y=
z=
22.1 Overview and Some Types of Rigid Motions
1. "Figure X is congruent to Figure Y." What does that statement mean?
2. Define the term, rigid motion.
3. Name three types of rigid motions, using "adult" terms.
Shapes and Measurement
Test-Bank Items
page 41
4. Name the type of rigid motion involved in each of the listed motions. Take into
account the different types of "eyes."
A
D
C
A
B
A
C
A
D
B
5. Name the single rigid motion that gives the indicated transformation.
A
A --> B ___________________
B
A --> F ___________________
F
E
D
B --> C ___________________
C
B --> D ___________________
A --> E ___________________
6. For the figures below, what single type of motion would accomplish each? Use
adult-level vocabulary.
D
A
C
A ––> B ______________________
A ––> C ______________________
A ––> D ______________________
B ––> C ______________________
B
7. See below. The unlabeled figure at top left is the original figure. Each of the
others, labeled a-h, is the result of a rigid transformation of the original.
Determine if each is the result of a translation, rotation, reflection, or glide
reflection.
Shapes and Measurement
Test-Bank Items
page 42
a.
b.
c.
d.
e.
f.
g.
h.
a
original
b
c
f
(spacing insufficient here)
g
d
e
h
8. Complete each of the following motions, being careful to fit your answer to the
grid:
a. reflection
b. translation
c. glide-reflection
22.2 Finding Images for Rigid Motions, and
22.3 A Closer Look at Rigid Motions
1. Lost! Find the missing information. (Drawing is all right. If your drawing is
"off," write words to communicate your intent.)
a. Missing: The vector for this translation.
image
original
Shapes and Measurement
Test-Bank Items
page 43
b. Missing: The angle and direction for this rotation.
original
center
image
2. Using the tracing paper, accurately find the image of the given shape for the
rotation indicated. (Tear the sheet given to you in two.) Your image should be on
the same sheet as the original shape. Paper-clip the two sheets to this page when
you finish. More sheets are available if you make an error.
(Instructor: Drawing and angle [with vertex of angle = center of rotation] supplied
on one sheet of tracing paper)
3. Complete the chart:
Type of rigid motion
In addition to the original figure, what needs to be
given in order to do the rigid motion?
a. Reflection
a.
b. Rotation
b.
c. Translation
c.
d. Glide Reflection
d.
Shapes and Measurement
Test-Bank Items
page 44
4. Complete the chart:
Type of rigid motion
Clock orientation of image compared to original
a. Reflection
a.
same
reversed
b. Rotation
b.
same
reversed
c. Translation
c.
same
reversed
d. Glide Reflection
d.
same
reversed
5. For each of the following pairs of figures, determine which single-step
transformation might take the shaded figure to the other. If a rotation, find the
center and angle of rotation. If a refection, show the line of reflection. If a
translation, give an arrow (vector).
a.a.
b.b.
a.
I love
I love
I love
Math.
Math.
Math.
a. __________________
6.
c. c.
b.
c.
I love
I love
I love
Math.
Math.
Math.
b. __________________
c. __________________
Draw accurately each of the following.
Shapes and Measurement
Test-Bank Items
page 45
image
A. the vector for this
translation
original
B. the line of
reflection involved
here
P
Q
R'
S
Q'
R
S'
P'
22.4 Compositions of Rigid Motions, and
22.5 Transformations and Earlier Topics
1. Complete to make true statements.
a. The composition of 10 different reflections and 5 different rotations could
possibly be a _______________________ or a ______________________.
b. A rotation always has ____________ fixed point(s). (How many?)
c. A translation with vector 50 cm long can be expressed by the composition of
two reflections, where the reflecting lines are (i) ________________ cm apart,
, (ii) ____________________________, and (iii)
_________________________
d. If P' is the image of P for a clockwise rotation of 140˚ with center X, then key
relationships tell us that _____________________ and
____________________.
e. The composition of 200 reflections could be achieved most efficiently with
____ reflection(s).
2.
(Instructor: To make the tests different, different shapes and locations were
drawn on the test copies.)
a. Use the MIRA to find the image of the given shape, for the composition
(reflect in line k) o (reflect in line x)
Shapes and Measurement
Test-Bank Items
page 46
k
x
b. Name the single rigid motion equal to the composition in part A, and give in
words or drawing all the important information about that motion (e.g., if
reflection, what is the line of reflection; if rotation, what are the center and
angle; if translation, ....).
________________________ with...
3. a. Give the final image of X for (translation, vector v) o (reflection in line m).
m
v
X
b. Name the single rigid motion equal to the composition in part A. You do NOT
have to give all the important information, just the type of motion.
(translation, vector v) o (reflection in line m) is a
_____________________
Shapes and Measurement
Test-Bank Items
page 47
4. a. Give an argument which shows this statement is true (write full sentences):
Every rigid motion can be expressed with at most three reflections.
b. Illustrate the statement given in part A, by showing the line(s) of reflection that,
with composition if necessary, give the image below.
original
image
5. TAKE-HOME PROBLEM. Due:
Name ____________________
Hand in this sheet, along with any scratch work. Note the pledge.
PLEDGE I did my own work without help, and I did not help anyone else in any
fashion. It was all right to use my class notes and the course materials.
Signed __________________________
Experiment with the composition of three reflections in three different parallel
lines, to predict what single motion the composition is. Describe the single
motion as completely as possible, relating the description to the relative
positions of the original lines. (This means if you think the composition is a
reflection, tell where the line of reflection will be with respect to the original
three lines; if you think the composition is a rotation, tell where the center will
be and the angle and direction; if you think the composition is a translation, etc.)
Answer: A composition of three reflections in parallel lines will be a
_________,
Shapes and Measurement
Test-Bank Items
page 48
where...
(The following questions are from a final examination in a course with about 1/3 of
the time spent on transformation geometry at the beginning of the course, so that
class had gone into transformation geometry to a greater depth.)
6. Completion.
a. The composition of 15 reflections in different lines with 13 rotations with
different centers might be a ____________________ or a
_____________________.
b. A reflection has _______________ fixed point(s). (how many?)
c. If two polygons are similar with a scale factor of 2, then a 20˚ angle in the
smaller polygon will correspond to an angle of ______ degrees in the larger
polygon.
7. a. Show the image of shape S, for the composition given by
(reflection in m) ˚ (translation with vector v)
m
v
S
b. What single rigid motion accomplishes the same thing as the composition in
part
a? ____________________
8.
Two lines intersect at point Q, making an angle of 60˚. What single rigid
motion equals the composition of reflections in those two lines?
________________?
Shapes and Measurement
Test-Bank Items
page 49
Describe this single rigid motion as completely as you can, from the given
information.
9.
Which of the following would be more valuable, if you could have only one of
them (and composition)? (Hint: Theory)
A. all possible rotations and translations
B. all possible reflections
____ because
10. Name the kind(s) of symmetry this figure has (the dots mean it continues
forever).
11. Joe and John were arguing about glide reflections. Joe says that the order that
you do the steps of a glide reflection makes a difference in your answer. John
says that it doesn’t.
Joe’s example:
Shape X is the answer if Joe translates first.
X
Shape Y is the answer if Joe reflects first.
Y
John’s example:
Shapes and Measurement
Test-Bank Items
page 50
Shape Z is the answer both if John reflects first and
Z
if John translates it first.
Who is correct about whether the order matters? Why (what was done wrong)?
23.1 Key Ideas of Measurement
1. a. What does the term “measurement” mean?
b. What process is involved in directly measuring something?
2. Suppose I reported the length of a bookshelf to be 2.12 meters. What range would
you expect the actual length of the bookshelf to lie in? Explain your answer.
3. Susanna stated that the length of her desk is 135.2 cm. What range would you
expect the actual length of the desk to fall within?
4. A measurement reported to be “22 inches to the nearest quarter-inch” could be at
the largest:
A. just under 22 14 "
B. 22
C. just under 22 18 "
D. 21 43 "
(Instructor: #9 is also like #2-#4.)
5. In each part, without calculation tell whether x or 10 is larger. Explain your
choice.
unit I
unit III
unit II
a. x unit I’s = 10 unit II’s (Don't forget to explain.)
b. x unit II’s = 10 unit III’s (Don't forget to explain.)
6. Anthony tessellated a rectangular piece of paper with thirty-six 1 inch by 2 inch
rectangles. Jennifer tessellated a page the same size and shape as Anthony’s, but
with 2 inch by 4 inch rectangles. How many rectangles did Jennifer use?
Shapes and Measurement
Test-Bank Items
page 51
7. At a store you bought 2 pounds of apples and 1 12 pounds of grapes. If the store’s
scale measures to the nearest half pound, what is the range of possible actual
weights for the sum of both purchases of fruits?
8. Pick the most appropriate unit from A-C for measuring each characteristic listed
below.
_____ A sector of a circle.
A. centimeters
_____ How much of an apple you ate.
B. cubic centimeters
_____ The length of the pencil you are using.
centimeters
C. square
_____ How far your 2 year old threw the ball.
9. Measuring to the nearest half-inch, what distances would be acceptable as 2 12
inches on a ruler? Sketch a part of a ruler showing your solution.
10. Give two different "key ideas" of measurement.
23.2 Length and Angle Size
1.
Find the measure of angle A: (Instructor: Include some angle; 3 are below.)
A
A
A
2. How many decimeters (to the nearest decimeter) are in 4.928 meters?
3. How many degrees of longitude separate 123 19 west longitude and 54 37 54
west longitude?
4. The minute hand on a clock goes through _____ degrees between 2:11 a.m and
3:12 a.m.
5. a. Using a compass and a straight edge, construct an angle that is 157.5. Do not
erase any important construction marks.
Shapes and Measurement
Test-Bank Items
page 52
b. Which key idea(s) of measurement did you use?
6. Take Home Extra Credit: (no working with others!)
Prove to me (i.e., convince me) that the 5 regular polyhedra in your kit of shapes
are the only 5 possible regular convex polyhedra.
7. Find the measure of an angle that is supplementary to an angle of size 35° 1’ 51”.
8. State TWO "key ideas" of measurement, with length or angle size in mind.
9. For the figure shown, with line AX parallel to line DY, find the following. For a-c
use the numbered angles ONLY.
B
a. Name a pair of vertical angles _____
25º
D
b. Name a pair of corresponding angles _____
1
c. Name a pair of alternate interior angles _____A
2
5
Y
4
80º
6
20º
d. Give the measure of angle 6 ______
3
C
X
10. The length of a baby's head (top of head to chin) is about one-fourth of the entire
length of a baby. If the baby is 56 cm long, how long is the baby's head?
A. 14 cm
11.
C. 224 cm
D. 18 cm
E. Not enough info
How many decimeters (to the nearest decimeter) are in 4.928 meters?
A. 1
12.
B. 70 cm
B. 5
C. 49
D. 9
E. 493
A male basketball player is most likely to be...
A. 2.2 dm tall
B. 2.2 m tall
C. 2.2 hm tall
D. 2.2 km tall
E. 2.2 cm tall
24.1 Area and Surface Area
(See 24.2 and 26.1 for items on surface area.)
1. Find the area of the trapezoid below using (a) the triangular region below as the
unit of measurement, (b) the square region below as the unit of measurement:
Shapes and Measurement
Test-Bank Items
page 53
Unit for part (a)
Unit for part (b)
a. _______________________
b. _______________________
2. There are __________ square decimeters in 9 square centimeters.
3. A child says: “I don’t get it! 1 yard = 3 feet, right? So why don’t 2 square yards
= 6 square feet?”
What would you explain to this child to help him or her understand? (Feel free to
draw pictures if that would be helpful.)
4. Draw a rectangular region on your paper that covers an estimated actual area of 8
square centimeters.
5. Juan, Kim, and Angelo each measured the area of a surface using the following
units of measure (shaded):
Juan's unit
Kim's unit
Angelo's unit
Which of the following could be a set of measurements they came up with?
A.
Juan: 60 units
Kim: 15 units
Angelo:
90 units
B.
Juan: 60 units
Kim: 240 units
Angelo:
40 units
C.
Juan: 40 units
Kim: 160 units
Angelo:
60 units
D.
Juan: 120 units
Kim: 30 units
Angelo:
80 units
6. Ollie measures the area of his kitchen floor and finds that it is 2400 square
decimeters. He needs to know how many square meters it is. Please help him by
Shapes and Measurement
Test-Bank Items
page 54
answering the question, and explain your response so that it will make sense to
him. A diagram may be helpful!
7. A 3 cm by 5 cm by 2 cm right rectangular prism is similar to a larger shape with a
scale factor of 2. What is the total surface area of the larger shape?
A. 60 cm2
B. 120 cm2
C. 124 cm2
D. 248 cm2
E. None of above
8. What is the area in square centimeters of the right triangle given below?
A. 60
B. 120
C. 240
26 cm
10 cm
D. 260
24 cm
E. None of the above
9. A Pringles can has a height of 30 cm and a diameter of 8 cm. The height of each
Pringle is 0.25 cm. If we assume that the Pringles are flat, what is the total
number of Pringles that will fit in the can?
A.
8
Pringles
3
B. 120 Pringles
C. 60 Pringles
D. 8 Pringles
10. Four gallons of paint were required to paint a scale model of a jet. The actual jet,
required 100 gallons of paint to be painted. What scale factor is associated with
changing all of the dimensions of the actual jet?
11. Determine the area of the shaded region on the given geoboard, if the unit of
measurement is 1 cm2. Use correct units.
1 cm2
Area = _____________
12. Find the exact area of the figures below. Show enough work that your strategy is
clear.
Shapes and Measurement
Test-Bank Items
page 55
Area = _________
Area = ________
13. a. Find the area of the polygonal
region on the geoboard (use
the natural unit).
Area = _____________
b. Which "key idea(s) of measurement" did you use in part a?
14. Ollie measures the area of his kitchen floor and finds that it is 2400 square
decimeters. How many square meters is it?
A. 2.4 m2
15.
B. 24 m2
C. 240 m2
D. not enough information
What is the area of the figure below (use the natural unit)?
A.
B.
C.
D.
E.
35 square units
30 square units
30 12 square units
32 square units
None of A-D
24.2 Volume
1. State an appropriate unit for measuring . . .
a. The width of an airplane runway: _______________________
Shapes and Measurement
Test-Bank Items
page 56
b. The size of a bathtub: _______________________
c. The size of a painting: ________________________
d. The arc that the tip of a clock hand traces out as it moves from one number to
another: __________________
2. Seven cubic meters have the same volume as __________ dm3.
3. For each statement, circle the most appropriate measurement:
a. The area of one page in a telephone book is about ... 6 cm
2
6 dm
2
b. The floor area in a small classroom is about ...
80 mm 80 dm
c. The amount of water in the lake is about ...
14 km
d. The volume of a slice of bread is about ....
mm
102 cm
3
14 km
102 dm
3
2
2
2
6 mm
80 m
2
2
14 km
3
102
3
3'. For each statement, circle the most appropriate measurement:
6 cm2
a. The area of this page is about...
6 dm2
6 mm2
b. The area of the floor in a math classroom is about...80 cm2 80 dm2 80 m2
c. The volume of a trash can in a classroom is about.. .45 cm3 45 dm3 45 m3
d.
The amount of water in an aquarium could be... 50 dm
50 dm2 50 dm3
4. Fred used 8 gallons of paint to paint a model of his jet. The actual jet is similar to
the model, but five times as long. How many gallons of paint will Fred need to
paint his jet?
5. If one cube has an edge of 2 cm and another has an edge of 3 cm, then the ratio of
their volumes is_____.
6. Convert the following:
a. 5 g = ___________ mg
Work:
b. 27 cm2 = ___________ m2
Work:
c. 1 m3 = ___________ cm3
Work:
7. Explain how could you illustrate to a child that there are 1,000,000 cm3 in 1 m3?
Shapes and Measurement
Test-Bank Items
page 57
8. Complete the following, using your "metric sense" as much as possible.
a. 2.7 kg = __________ g
b. 15.3 m = __________cm
c. 12 mm = _____cm
d. 2153 g = __________kg
e. 2.3 mm2 = _________ cm2
f. 0.4 m2 = _____cm2
g. 2257 cm3 = _________ dm3
h. 0.4 m3 = _________cm3
i. 232 m3 = ___________L
j. 300 mL = _____cm3
9. How is area different from volume? Can anything have area without having
volume? Can anything have volume without having area? Give an example.
10. How is surface area different from volume? Your explanation may include an
example, but make clear how the ideas, not just the numbers, are different.
11. Give the surface area and the volume of the 3D shape shown. Use the natural
units. (Instructor: Choice.)
Surface area = ___________
Volume = __________
Shapes and Measurement
Test-Bank Items
page 58
Surface area = ___________
Volume = __________
12. Cube A has edges that are 3 times as long as the edges of Cube B. The volume of
Cube A is 108 cubic cm. What is the volume of Cube B?
A. 36 cm3
B. 324 cm3
C.12 cm3
D. 4 cm3
E. None of A-D
25 Counting Units Fast: Measurement Formulas
1. You would like to buy wheels for your child's go-cart so that the wheels will roll 1
meter in one full revolution. About what should the diameter of each wheel be?
1'. You would like to buy wheels for your child's go-cart so that the wheels will roll 1
meter in full revolution. About what should the diameter of each wheel be?
A. 0.32 m
B. 0.56 m
C. 1.13 m
D.1 m
E. 3.14 m
2. What is the area of the trapezoid shown below?
5.5 cm
3 cm
2 cm
8.5 cm
3. What is the area of a rectangle similar to the one shown with a size change of
scale factor 4?
Area=5 square units
Shapes and Measurement
Test-Bank Items
page 59
4. What is the area of a rectangle with perimeter 20 meters and base 6 meters?
4'. What is the area of a rectangle with perimeter 20 meters and base 6 meters?
A. 120 m2
B. 60 m2
C. 160 m2
D. 24 m2
e. 84 m2
5. Explain why a formula for the volume of a prism is the following (use a drawing
if that is helpful):
Vprism=(Area of the base)(height)
6. Find the volume of the cone with the following dimensions:
7 cm
7.5 cm
5 cm.
6'. Find the approximate volume of the cone with the following dimensions:
7 cm
7.5 cm
A.
B.
C.
D.
E.
137.4 cubic cm
45.8 cubic cm
147.3 cubic cm
69.1 cubic cm
183.3 cubic cm
5 cm
Shapes and Measurement
Test-Bank Items
page 60
7. Find an expression for
the area of the figure to
the right, in terms of the
indicated measurements.
Angles that look like
right angles are right
angles.
c
a
(Instructor: alternative
versions below.)
n
7'. Find the area of the figure, if c = 8 cm, n = 7 cm, and a = 20 cm. (Angles that
look like right angles are right angles.)
7". Find the area of the figure below (angles that look like right angles are right
angles).
Shapes and Measurement
Test-Bank Items
page 61
A. na + 12 (nc) square units
B. n(c – a) + 12 (nc) square units
c
C. n(c + a) +
1
2
1
2
(nc) square units
D. n(a – c) + (nc) square units
E. None of A-D
a
n
8. Explain why a formula for the area of a trapezoid is:
Atrap= 12 (Base1  Base2 )(height) .
Base1
height
Base 2
9. The lengths of the sides of the triangle below are x, y, and z, and with p as shown.
Derive a formula for the area of the triangle, without introducing any new
variables. Write enough to make your thinking clear.
y
p
x
z
10. A regular square pyramid has 2 cm edges on its base, and the other edges are 3
cm long.
a. Sketch a net for a pyramid that is similar to the one described above, with a
scale factor of 1.5. Make sure to label the lengths on your sketch.
Shapes and Measurement
Test-Bank Items
page 62
b. If the original pyramid has a volume of 3.5 cm3, what will be the volume of the
new pyramid in part a? Show (and explain) your work.
11. An ice cream cone filled just to the top holds 28 cm3 and has a height of 6 cm.
What is the area of the base of the cone?
A.
14
3
cm2
B. 2 14 cm2
C.
14

cm2
D. 14 cm2
12. A cylindrical container has a volume of 320 cm3 of water, and the diameter of
the container is 8 cm. What is the height of the container?
13. Find the area of the triangle.
3 cm
4 cm
14. What is the AREA of the figure below? (Don’t forget the units.)
2 cm
Area = ______
6 cm
5 cm
15. To the nearest inch, a rectangle measures 2 inches by 4 inches. The area of the
rectangle could be anywhere between _______ square inches and _________
square inches. Please express your answer(s) as fractions or mixed numbers.
16. If the diameter of a tennis ball is a little more than 6 cm, about what are the
surface area and volume of the tennis ball?
Surface Area:_______
Volume:_______
17. a. Sketch the 3D shape that has the net shown. (The middle part is a rectangle.)
Shapes and Measurement
Test-Bank Items
page 63
4 cm
6 cm
4 cm
b. Name the 3D shape as fully as possible. _____________________________
c. Give the volume and surface area of the 3D shape. (Use correct units. Show
your work for credit.)
Volume = ________________
Surface area = __________________
18. Bonus: Explain how you can derive the area formula of a circle using triangles.
OR Explain how you can derive the area formula of a trapezoid using
parallelograms.
For either: In your explanation, use labels from your drawings as variables for
your formulas.
19. The area of the triangle below is 24 square units. Each of the sides in a triangle
can be used as a base. For each the triangles shown, choose a different side as the
base and then draw the corresponding height for that base. Find the measurement
of that height.
6
13
6
9
13
9
6
13
9
20. What is the area of the circle below?
Shapes and Measurement
Test-Bank Items
page 64
length of diameter=
3 inches
21.
A.
B.
C.
D.
E.
2.25π square inches
9π square inches
6π square inches
3π square inches
1.5π square inches
What is the volume of the box below?
A.
B.
C.
D.
E.
Height of box = 2.5 inches
Area of bottom of box = 35 sq. inches
Length of diagonal of bottom = 8.5 inches
87.5 cubic inches
743.75 cubic inches
297.5 cubic inches
46 cubic inches
37.5 cubic inches
22. If the volume of a right circular cylinder with radius 5 cm is about 550 cm3, about
what is the height of the cylinder?
A. 110 cm
B. 35 cm
C. 22 cm
D. 7 cm
E. 3 cm
23. The perimeter of a square is 20 cm. What is its area, in cm2?
A. 400
B. 80
C. 25
24. What is the area of the triangle to the right, in
cm2? The given measurements are in cm.
A. 32
B. 31
C. 24
D. 16
E. None of A-D
D. 20
4
E. None of A-D
6
13
8
25. Give the area and the perimeter of the following shape. The curved edges are
semicircles with the indicated points as centers. Show your work.
Shapes and Measurement
Test-Bank Items
page 65
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Area =
Perimeter =
26. In the shape below, ABCDE is a regular pentagon, M is the midpoint of side AB,
and A is the center of the circular arc.
A
M
E
4 cm
B
D
C
Area of the shaded shape =
Perimeter of outer shape =
27. In the shape below, the curves are semicircles, with the centers and radii
indicated.
radius 5 cm
radius 3 cm
(slightly easier version for perimeter)
Shapes and Measurement
Test-Bank Items
page 66
radius 5 cm
radius 3 cm
13 cm
Area =
Perimeter =
26.1 Pythagorean Theorem
1. Find the length of diagonal AC in the following right rectangular prism:
A
?
8 cm
5 cm
12 cm
C
1'. Find the length of diagonal AC in the following right rectangular prism.
Shapes and Measurement
Test-Bank Items
page 67
A
?
8 cm
5 cm
C
12 cm
A. 15.3 cm
B. 13 cm
C. 9.4 cm
D. 20 cm
E. 48 cm
2. A plan calls for running a telephone line from P to Q to R. The line costs
$7/meter. If it is feasible to run the line directly from P to R, how much money
would one save?
R
9m
P
Q
12 m
2'. A plan calls for running a telephone line from P to Q to R. The line costs
$7/meter. If it is feasible to run the line directly from P to R, how much money
would one save?
R
9m
P
12 m
A.
B.
C.
D.
E.
$ 42
$105
$147
$ 6
$378
Q
3. What is the length in centimeters of the base of a rectangle with diagonal of
length m cm and width of length n cm?
4. Find the perimeter and the area of the isosceles triangle below. Make sure to show
your work. The drawing is not to scale. (Hint: the triangle can be cut into two
congruent triangles.)
Shapes and Measurement
Test-Bank Items
page 68
perimeter: ________________
13 cm
13 cm
area: ____________________
10 cm
5. Frank enlarged the dimensions of the trapezoid below by a scale factor of 150%.
15 cm
12 cm
20 cm
9 cm
Find the following information:
a. What is the perimeter of the original figure?
b. What is the area of the original figure?
c. What is the perimeter of the new figure?
d. What is the area of the new figure?
6. What is the area of the right triangle shown to the
right?
4 cm
5 cm
A. 6 cm2
B. 10 cm2 C. 12 cm2
Shapes and Measurement
D. 20 cm2
Test-Bank Items
E. Not enough information
page 69
7. What are the perimeter and area of the given figure? Use correct units. (The
curved part is a semicircle.)
P = __________
12 cm
A = __________
8 cm
8m
8. Find the area of the figure to the right.
9m
7m
9m
Give the units. Show your work.
11 m
Area = _____________
15 m
9. What is the length of a diagonal of a square that is 12 cm on each side?
10. What does the Pythagorean theorem assert about a right triangle with legs of
lengths x cm and y cm?
11. Find the volume of a right square pyramid with a height of 8 cm and the base
having sides with length 12 cm. Use correct units. Show your work for credit.
Volume = ____________
12. An isosceles-triangular right prism has the length dimensions indicated. The
base of the isosceles triangle has length 12 dm.
Shapes and Measurement
Test-Bank Items
page 70
15 dm
12 dm
8 dm height
a. Find the missing lengths of the sides of the isosceles triangles.
b. Draw a net for the prism, giving the dimensions.
c. Give the volume and the surface area of the prism. Show your work.
Volume = _________________
Surface area = _______________
13. Find the area and perimeter of the shape below. The curved part is a semicircle.
Show your work for credit.
50 m
30 m
Area = _____________________
Perimeter = ______________________
14. What is the length in centimeters of the base of a rectangle with diagonal of
length m cm and height of length n cm?
A.
m2  n 2
B.
m2  n 2 C. m  n
15. What is the area of the hexagonal region to
the right, in square centimeters? Assume
that lines that look parallel are parallel, and
that angles that look like right angles are
right angles.
A. 100 B. 80 C. 65 D. 40 E. 29
D.
mn
2
E.
m n
2
4 cm
5 cm
5 cm
5 cm
10 cm
16. Find the area and the perimeter of the shape below. The indicated point is the
center of the arc.
Shapes and Measurement
Test-Bank Items
page 71
3 cm radius
10.9 cm
60Þ
8 cm
Area =
Perimeter =
26.2 Some Other Kinds of Measurements
1. a. State the "key ideas" of measurement.
b. One of the key ideas does not apply to all quantities. Give an example, making
clear why the key idea does not apply there.
2. Who had the highest average rate of speed? Show your work.
Ann, who went 450 miles in 7.5 hours.
Ben, who went 20 miles in 24 minutes.
Carlo, who went 175 miles in 3 hours.
Daniela, who went 36 miles in 32 minutes.
3. "My gpa for the first 90 units is 3.1. This semester, I'm taking 12 units and expect
to get a 3.5. If I do that well, what will my overall gpa be, after this semester?"
Your answer? (Show your work.)
Final question:
Shapes and Measurement
Test-Bank Items
page 72
In your opinion, what are the 3 most important things (ideas, formulas, concepts,
etc.) that you have learned and will take with you from this course?
Shapes and Measurement
Test-Bank Items
page 73
Reasoning about Shapes and Measurement
Answers for Test Bank Items
Answers may be given in only one of fraction/decimal form. You will know your stance
on the use of calculators and decimals.
16.1-16.2 Shoeboxes Have Faces and Nets; Introduction to Polyhedra
1. T The base is one face and has n sides. Each of these sides gives a (lateral) face.
1'. T There are n vertices on the base, plus the one shared by the lateral faces.
2. T From Euler’s formula, V + F = E + 2, if V and F are odd, then their sum is
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
even, but then if E is odd, E + 2 would be odd also.
a. octagonal
b. 18, 12, 8
c. 31, 31
d. 15, 10, 7
a. 6, 18, 12
b. 8, 5, 5
c. 14, 21
d. 17, 17
e. 12
An octagonal pyramid
102, because there are 2 bases, plus a face for each side of the 100-gonal base.
300, because there are 100 edges at each of the bases, plus a lateral edge for each
vertex of the 100-gonal base.
200, because there are 100 vertices at each of the two bases.
Yes—e.g., a cube
a. 9, 9
b. 9
c. 30
d. 11
D
D
B
D
E
a. 10
b. 71
Shapes and Measurement Page 74
3/9/16
c. n
d. 48
16.
17.
18.
e. triangular regions
a. 24, 16
b. n, n+1
c. 20
d. 150
e. 1440
For a polyhedron with V vertices, F faces, and E edges, V + F = E + 2.
A
C
19.
20.
21.
22.
23.
24.
25.
26.
27.
C
i. B ii. C
iii. D
i. A
ii. C
iii. C
E
D
D
D
A, B, and D give nets; C does not.
Mark according to your emphasis in class.
15.
16.3 Representing and Visualizing Polyhedra
1.
a. hexagonal pyramid
b. right hexagonal prism
c. right square prism
d. (right) trapezoidal prism
e. oblique octagonal prism
f. (right) triangular right prism
g. (right) regular octagonal prism
2.
3.
4.
h. (oblique) pentagonal prism
a. triangular right prism
b. rectangular pyramid
a. Drawing should show parallel pentagonal bases and some indication of right
angles, along with hidden edges.
b. Drawing of triangular pyramid
(Correct) drawings for two different nets for a cube.
Shapes and Measurement Page 75
3/9/16
5.
6.
2 (base, plus 1 hidden face)
6'. 3 (base, plus 2 hidden faces)
6". Look for drawing of pentagonal pyramid, with hidden edges. Response should
7.
8.
include "right"/"oblique," as appropriate
Drawing should show a right equilateral triangular prism, perhaps with hidden
lines.
Left 3D figure
Right 3D figure
front
right
top
front
right
top
9. Answers will depend on the particular shapes X and Y
10. a. 51, with explanation (e.g., 1 vertex at the top, plus the 50 at the base)
b. 100, with explanation
16.4 Congruent Polyhedra
1.
2.
Choices c and d are congruent.
a.
b. A shape congruent to part a but different view.
16.5 Some Special Polyhedra
1.
2.
a. 5
b. (regular) tetrahedron, hexahedron, octahedron, dodecahedron, icosahedron
A cube is a regular polyhedron because all of its faces are squares (regular
quadrilaterals) and all of the arrangements at each vertex are the same
Shapes and Measurement Page 76
3/9/16
3.
4.
5.
A regular polyhedron is a polyhedron all of whose faces are congruent regular
polygonal regions of one kind, with the same arrangement of regions at each
vertex of the polyhedron.
a. T
b. F The regular polyhedra were known to the ancient Greeks, BC.
c. T
d. F There are four other types of regular polyhedra
e. T
Two of these…
regular tetrahedron
triangle
4 vertices
6 edges
4 faces
regular hexahedron
square
8 vertices
12 edges
6 faces
regular octahedron
triangle
6 vertices
12 edges
8 faces
regular dodecahedron pentagon
20 vertices
30 edges
12 faces
regular icosahedron triangle
12 vertices
30 edges
20
faces
17.1-17.3 Review of Polygon Vocabulary; Organizing Shapes;
Triangles and Quadrilaterals
1.
Quadrilaterals
Kites
Trapezoids
Isosceles trapezoids
Parallelograms
Rectangles
Rhombuses
Squares
2.
a. Sketch of right isosceles triangle or obtuse isosceles triangle
b. Sketch of a trapezoid that is not isosceles (two right angles)
c. Sketch of a square
d. Sketch of a prism with right-triangular bases (probably acceptable, unless you
have emphasized the former, or a right prism with triangular bases)
3.
a. Not possible, because the parallel sides will force at least two right angles (if
there is one)
b. Sketch of any rectangle/square
c. Sketch, exactly one right angle, 5 sides
d. Sketch of isosceles obtuse triangle
e. Not possible, because every rhombus is a special kite
360˚. A pair of interior-exterior angles at each vertex will give 180˚, or 180n˚ in
all. Subtract (n – 2)180˚ for the interior angles: 180n – (n – 2)180 = 360.
4.
Shapes and Measurement Page 77
3/9/16
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
a. Always true, because of the rotational symmetry (or whatever reason fits your
class discussion)
b. Sometimes true, when the kite is a rhombus
a. Sometimes true, when the parallelogram is a rectangle
b. Always true, because of the hierarchical relationship
c. Sometimes true, because a scalene triangle may be right or obtuse as well
Yes. A sketch of a parallelogram/rectangle/rhombus/square shows that Definition
A is all right, but because the shape has both pairs of opposite sides parallel,
Definition B is not satisfied.
T, T, T, T, F, F, T, F
It is not possible, because if there is one right angle, the parallel sides will assure
that there is a second right angle.
Any non-rhombus kite
a. square
b. Not possible, because all the sides of a rhombus have the same length
c. Not possible, because the equilateral condition would make the shape a
rhombus, but the isosceles trapezoid condition would make it a square, violating
the “not regular” condition.
a. F b. T c. F d. F e. T f. T g. F
a. square
b. rectangle
a. Sketch of trapezoid with 2 right angles (not a rectangle)
b. Sketch of a square, but an “ordinary” kite can have equal diagonals, so long as
one is along the perpendicular bisector of the other.
True example: Sketch of any rectangle/square.
Counterexample: Sketch of non-square rhombus or non-rectangular parallelogram
D
A drawing might involve an optical illusion or a special case.
What seems to be true for even several examples may break down on an
unexamined example.
Shapes and Measurement Page 78
3/9/16
19.
quadrilateral
trapezoid
kite
parallelogram
rhombus
isosceles trapezoid
rectangle
square
19'. Polygon to kite and trapezoid, then trapezoid to rectangle and then to square, with
kite to square also.
20. a. 144 degrees
b. 72
c. 0
d. 104
21. a. not possible (one right angle in a parallelogram assures that there are 4)
b. drawing of isosceles right triangle
c. not possible (every rectangle is a special parallelogram)
d. drawing of a rhombus that is not a square
e. drawing of a concave hexagon ("dented in" in at some vertex)
22. a. T
b. F (drawing of non-square rectangle)
c. T
d. T
e. F (drawing of some non-parallelogram trapezoid)
f. T
g. F (drawing to show violation of some rectangle characteristic)
h. F (drawing to rhombus with unequal lengths for the diagonals)
i. T
j. T
23. The student may be assuming that a rhombus must not have right angles. Ask her
what a rhombus is, and whether a square meets the conditions.
24. a. 5
b. 5,150 (Look for work.)
25. D
Shapes and Measurement Page 79
3/9/16
26. A
27. E
28.
29.
30.
31.
D
a. 100
b. 60
c. 20
2700 (Look for work.)
a. ST drawings: rectangle
non-rectangular parallelogram
b. AT
c. ST drawings: square
non-square rhombus
d. NT
32. The student's reasoning is incorrect, because he/she overlooked that the 360˚ in the
angles in the middle of the hexagon do not contribute to the angle sum for the
hexagon. Subtracting that 360˚ from the 1080˚ would give the correct sum, 720˚.
33. a. drawing of a rhombus; "rhombus"
b. Not possible. An equilateral triangle has 60˚ angles, none of which is obtuse.
c. drawing of pentagon with 3 right angles (e.g., "house" with right angle at top)
polygons
34.
polygon
quadrilaterals
or
quadrilateral
rectangle
rhombi
rect.
sq
rhombus
square
35. a. A few minutes work gave 9 different shapes, but that is unchecked. A reflected
shape is not counted as different.
b. One strategy is to build up on each of the two different shapes possible with two
rhombuses (2 in a row, and a chevron).
36.
37.
38.
39.
Ten dots can be arranged to form a triangle, as with bowling pins.
A 30˚-30˚-120˚ meets the criteria (obtuse, isosceles).
All the sides of a rhombus are the same length.
E.g.--Each of the angles of a rectangle is a right angle. Also, the diagonals of a
rectangle are the same length.
40. (2 facts) The diagonals of a rhombus are perpendicular, bisect each other, and
bisect the angles of the rhombus.
Shapes and Measurement Page 80
3/9/16
41. (2 facts) The diagonals of a rectangle are the same length, bisect each other, and
cut the rectangular region into 2 (singly) or 4 parts of the same size.
42. (2 facts) Opposite sides are parallel and have the same length. Opposite angles are
the same size. Diagonals bisect each other. Each diagonal cuts the region into two
congruent pieces.
43. B (when the parallelogram is a rectangle)
44. A
45. E
46. B
47. E
48. A
49.
50.
51.
52.
53.
54.
55.
T, T, T, T, F, F, T, F
D
e, d, a, b, a, d, a, e
From (n – 2)180 = 180,000, one finds that n = 1002 gives such a polygon.
160˚
B
D
18.1 Symmetry of Shapes in a Plane
1. 8, 8
2-3. Check for the type of symmetry, and the line/center specified.
4. a. 12
b. 30˚
5. 1, 0
6. a. 90
7.
8.
9.
10.
11.
12.
13.
b. 90
c. 4
F (360 ignores all the rotations involving a part of a degree)
The drawing should show an equilateral triangle with 3 reflection lines.
The drawing should have a reflection symmetry, with the line shown.
The drawing should have a rotational symmetry, with the center highlighted.
6 lines of symmetry; drawing should show them
a. 72˚ b. 360˚
c. 90˚
a. 5 lines of symmetry should be drawn
b. 1 line of symmetry should be drawn
c. 1 (vertical) line of symmetry should be drawn
Shapes and Measurement Page 81
3/9/16
18.2 Symmetry of Polyhedra
1.
2.
3.
4.
7 reflection symmetries; 12 rotational symmetries (the 60˚ ones, plus 6 from axes
perpendicular to the 3 pairs of opposite lateral edges and from the 3 pairs of
opposite lateral faces
a. 3
b. 4
F (There are 3 planes giving reflection symmetries.)
5 reflection symmetries, 8 rotational symmetries (5 axes, 1 giving 4 rotational
symmetries including the 360˚ one, the other 4 giving a 180˚ rotational
symmetry; the 360˚ for them won’t be new. Mark this according to how strictly
you have counted; the fact that where the axis is for a 360˚ rotational symmetry
does not matter is subtle).
19.1 Tessellating the Plane
1.
a. You will likely get a drawing.
b. You will likely get a drawing—check that there is a clear, repeatable pattern.
c. Each angle of a regular pentagon has size 108˚, and 108˚ angles will not fit
together to make 360˚.
2.
a. no
b. yes
c. yes
d. yes
Regular octagons have angles with 145˚, and angles of 145˚ will not fit together to
make 360˚.
a. no
b. yes
c. yes
3.
4.
5.
6.
7.
yes, yes, no, no
Only b, c, and e can tessellate the plane.
Regular pentagons cannot tessellate the plane, because each angle of a regular
pentagon is 108˚, and such angles cannot fill in around a vertex.
8.
There should be enough work shown that it is clear that the student could go
beyond 10 copies.
19.2 Tessellating Space
Shapes and Measurement Page 82
3/9/16
1.
a and d could tessellate space.
20.1-20.2 Size Changes in Planar Figures; More about Similar Figures
1.
2.
3.
4.
5.
They are similar, with scale factor 1.6 (or 0.625).
a. 1 ft. ––> 3 ft., but Fredrick forgot to measure the 3 ft. from the center.
b. He should have measured the new distances from the center, else the resulting
shape will not be similar to the original.
Yes. Since the triangles are equilateral, all three pairs of corresponding angles will
have the same size (60˚). This fact assures that the 8:12 ratios are all equal, but
that fact can be ascertained directly, knowing that the lengths of the sides of an
equilateral triangle are all the same.
a. 1 13 (or equivalent)
b. 13
a. 2 115
b. 66 23
6.
7.
c. 35
F
a. Each has two pairs of angles the same size—the 98˚ ones and the 36˚ ones.
61
 3.389 cm
b. x = 18
y = 9.9 cm
8.
Other “fixes” of the False ones are possible.
a. False (…3.25 times as long as 2 cm, OR …is 2.25 times longer than 2 cm)
b. True
c. True
d. False (…is about 170% more than 1.3, OR …is about 270% of 1.3)
e. False (…times as long as 2 inches, OR …is 3 14 times longer than 2 inches)
f. False (…an increase of 20%)
g. True
h. False (Possibly: If two rectangles have lengths of sides in the same ratio, then
they are similar.)
9.
i. True
BC = 21 23 cm
m BCA  35Þ
m D  85Þ
10. a. 3
b. 3 13 cm by 4 cm
DE = 6 23 cm
m ECD  35Þ
m E  60Þ
c. Yes, because size changes give similar shapes.
11. Sketches should show….twice the length of the original, 3 times the length of the
original, and 4 times the length of the original
Shapes and Measurement Page 83
3/9/16
12. a. 50
b. 60
c. 12
13. Scale factor =
8
5
or 1.6;
x = 6 14 or 6.25 cm
14. 525 miles. Look for explanation. (The SD-LV distance on map A is irrelevant.)
15. a. DF = 11 23 cm, EF = 8 13 cm
b. About 41 23 cm2
16. a. 22.50
b. 36
c. 32
17. a. Sketch should show segment with length 3 times that of the original.
b. Sketch should show segment with length 3 12 times that of the original.
c. Sketch should show segment with length 2 14 times that of the original.
18. a. Perhaps not, because corresponding angles may not be the same size, for any
correspondence.
b. Will be similar, because all angles have 90˚ and the ratios of the lengths of
corresponding sides will all be the same, since the sides of a square are all the
same size.
19. 200 cm2
20. Scale factor = 125 (since it is a shrink); x = 2.5 cm
21. Check to see that the distances from the center to the image points are only twice
those from the center to the original points. Visually, the sides of the image
quadrilateral should be parallel to the sides of the original.
22. y = 100˚; x = 15 (scale factor is 2.5)
23. a. Two pairs of angles have the same sizes.
b. x = 3; y = 15
24. B
25. A
26. D
27. E
28. No, since some ratios between corresponding lengths are different,
29.
30.
31.
32.
7
4
and
10
7
.
B
B
B
Look for both checking (at least three) pairs of angles for being the same size, and
for checking that the four ratios, given length: unseen length, all have the same
value.
Shapes and Measurement Page 84
3/9/16
33. No, because 6:9 ≠ 4:6.2 (even though all four pairs of angles are the same sizes and
6:9 = 8:12).
34. a. 17.5, 25, 12.5 (any order)
b. 90, 90, 127, 53 (any order)
20.3 Size Changes in Space Figures
1.
2.
3.
x = 2.9 inches
y = 4.7 inches
The pyramids are not similar, because 7:4 ≠ 10:7.
a. 4
b. 14
4.
2.32  (6  22 )  126.96 cm2
5.
87.8 cubes
Alternatives, in order: 109.8; 109.8; 131.7
cm3
6.
4
7.
a. S.A. = 66 sq. units Volume = 36 cubes
b. S.A. = 9  66  594 sq. units
Volume = 27  36  972 cubes
a. 8 cu. inches
b. 49
8.
9. 40 500 gallons
10. a. Not necessarily, because length dimensions may give different ratios.
b. Yes, because each of the face angles is 90˚, and the ratios of the lengths of
corresponding edges will necessarily be equal, since all the edges of given cube
are the same size.
c. Not necessarily. Although the bases will be similar, there is no assurance that
the lateral edges will have lengths in that same ratio.
11. C
12. 4 x (scale factor)3 cannot exceed 600, or (scale factor)3 cannot exceed 150. The
scale factor can be 5. (With the cubes to be used, only whole numbers as scale
factors are possible.)
13. 5
14. A
21 Curves, Constructions, and Curved Surfaces
2.
3.
Check for drawings meeting the described figures.
cone
4.
a. never true, by definition
b. sometimes true, if it is a right circular cylinder
Infinitely many (e.g., any plane through the centers of the bases, i.e., any plane
containing the line through the two centers)
5.
Shapes and Measurement Page 85
3/9/16
6.
7.
8.
a. F because a chord does not have to go through the center of the circle.
b. T because no point is more than the radius’ length from the center, so no two
points can be more than two radii’s length in total.
c. F because the ratios of their lengths and widths do not have to be equal—e.g., as
with a 1-by-8 and a 2-by-4.
d. T because corresponding angles are the same size since they are similar, and
since the scale factor is 1, corresponding lengths must be equal.
Look for proper construction marks. You can also use a compass to check for the
location of centers and the correct radii.
a. T
b. F
c. F (The ratio of the volumes would be 278 .)
9.
minor arc LM or MP or NP or NL
major arc LNP or MPL or PNM or NLM
diameter LP
Drawing should include chord AB
Drawing should have a sector shaded in.
10. always true. A diameter is a line segment that passes through the center of the
circle and has its endpoints on the circle, making it a chord.
11. a. Look for marks to show the copying.
12.
13.
14.
15.
b. Look for marks to show the perpendicular bisector construction.
a. Look for copy-angle construction marks.
b. Look for bisect-angle construction marks.
Look for marks that would assure a rectangle. These might not reflect the official
definition of rectangle as a parallelogram with a right angle, so the follow-up
discussion could pursue the other methods.
Look for marks involving the construction of a perpendicular, the bisection of a
right angle, and the joining of a 90˚ angle and a 45˚ angles.
Look for marks involving the bisection of the angle, and the joining of the angle or
a copy with one of the half-angles.
16. Look for marks assuring parallelism.
17. x = 100˚
y = 50˚
z = 40˚
22.1 Overview and Some Types of Rigid Motions
1.
2.
One of the figures is the image of the other for some rigid motion.
A rigid motion is a movement that does not change lengths or angle sizes. (The "or
angle sizes is actually redundant; a student with good background in
transformation geometry may know that and state only the lengths criterion.)
Shapes and Measurement Page 86
3/9/16
3.
4.
reflection (in a line), translation, rotation (about a point)
rotation; reflection; translation
5.
6.
7.
translation, glide-reflection, rotation, reflection, rotation
reflection, translation, glide-reflection, glide-reflection
a. translation b. reflection c. reflection d. translation
e. rotation
f. reflection g. rotation
h. rotation
a.
b.
8.
(any order
c.
22.2 Finding Images for Rigid Motions, and
22.3 A Closer Look at Rigid Motions
1. a. Vector from any point in the original to the corresponding point in the image
b. Angle defined by any point in original to center to corresponding point in image.
Direction—ccw for acute angle (or cw for large angle)
2. A visual check will expose mistakes.
3.
4.
5.
6.
a. the line of reflection
b. the center, and the angle of rotation (including clock direction)
c. the vector (or the distance and direction of the translation)
d. the vector for the translation part, and the (parallel) line of reflection
a. reversed
b. same
c. same
d. reversed
a. rotation, center where the lightning-bolts meet, angle 90˚ counter-clockwise
b. translation, vector by arrow from shaded figure to corresponding point on other
c. reflection, with line of reflection horizontal, halfway between the two banners
A. Join any point in the original to its correspondent in the image.
B. The line through the two points in which the quadrilaterals intersect.
22.4 Compositions of Rigid Motions, and
22.5 Transformations and Earlier Topics
1. a. rotation, translation (either order)
b. one
c. 25, parallel, perpendicular to the line of the vector
Shapes and Measurement Page 87
3/9/16
d. mPXP '  140Þ and mPX '  mPX (either order)
e. 2
2. b. Rotation, center where k and x intersect, angle and direction indicated (about twice
an angle formed by k and x, directed from x toward k through the angle doubled).
3.
5. a.
X
final
image
b. glide-reflection
4. a. Every rigid motion is one of these: reflection, translation, rotation, or glidereflection. A translation or rotation can be achieved by the composition of two
reflections. A glide-reflection can by achieved by three—two for the translation,
plus the reflection of the glide-reflection.
b.
original
Heavy points are midpoints, and give the line
of reflection (heavy).
Join any point on the
dashed figure to its
corresponding point
on the image, to get
the vector.
image
Shapes and Measurement Page 88
3/9/16
5.
6.
A single reflection. Its line of reflection is describable, but difficult to do in
generality. Suppose the lines are vertical, with the order of the composition from
right to left. If the distance x between the second and third reflection lines is
greater than the distance y between the first and second reflection lines, then the
single reflection line will be x to the left of the first line of reflection (or y to the
right of the third line of reflection).
a. reflection, glide-reflection (any order)
b. infinitely many
c. 20
7.
15. a.
m
v
S
final
image
8.
9.
b. glide-reflection
Rotation, center Q, 120˚ (can’t tell whether cw or ccw)
B, because each of the other types of rigid motions can be achieved by
compositions of reflections. Choice A would not enable one to do reflections or
glide-reflections.
10. Many translations, rotations, reflections, and glide-reflections
11. Joe's example involves a vector and line of reflection that are NOT parallel, and
order does matter in such cases. John's example follows the usual definition of
glide-reflection, in which the vector and line of reflection are parallel. In those
cases, order does not matter.
23.1 Key Ideas of Measurement
Shapes and Measurement Page 89
3/9/16
1.
a. “Measurement” can refer to the process of assigning a value to a quantity, or to
the result of that process.
b. A direct measurement of a characteristic involves comparing the given object
with a unit having the same characteristic as the object, repeating as necessary so
that the two match on the characteristic
2.
≥ 2.115 meters and up to 2.125 meters.
3.
≥ 135.15 cm and up to 135.25 cm
4.
C
5.
a. x, because unit I is smaller than unit II, so it will take more of them to match 10
unit II’s.
b. 10, because unit III is smaller than unit II, so it will take more of them to match
x unit II’s.
6.
9
7.
≥ 3 14 pounds but < 3 43 pounds
8.
C, B, A, A
9. As low as 2.25 inches, and up to 2.75 inches. (Should have sketch of ruler.)
10. Any two (Measurement involves a matching with copies of a unit; standard units are
good for communication and permanence; measurements (besides counts) are
approximate; a whole can be measured by cutting it into parts, measuring them,
and adding those measurements.)
23.2 Length and Angle Size
1. The samples are 65˚, 125˚, and 70˚.
2. 49
3. 68˚ 22’ 25”
4. 366
5. a. Check for, for example, 90 + 45 + 22.5, with the 45 and 22.5 coming from
bisections.
b. The measurement of a whole is the sum of the measurements of its parts.
6. See the answer for text Exercise 28.b.
Shapes and Measurement Page 90
3/9/16
7. 144˚ 58' 9"
8. Any two, okay if focus is on length or angle size (Measurement involves a matching
with copies of a unit; standard units are good for communication and permanence;
measurements (besides counts) are approximate; a whole can be measured by
cutting it into parts, measuring them, and adding those measurements.)
9. a. 4 and 6 (either order)
b. 3 and 6 (either order)
c. 3 and 4 (either order)
d. 85˚
10. A
11. C
12. B
24.1 Area and Surface Area
(See 24.2
1. a. 6
b. 12
2. 0.09
3. A picture of an actual 2 square yards, marked in square feet might be the best answer.
4. Check for something in the neighborhood, unless you allowed a ruler.
5. B
6. 24m2, because 1 dm2 = 0.01 m2
7. D
8. B
9. B
10.
(Even though the paint is measure in volume units, the issue is the relative sizes
of the areas.)
1
5
11. 7 12 cm2
Shapes and Measurement Page 91
3/9/16
12. Left figure: 6 12 units
Right figure: 10 12 units
13. a. 17 12 units
b. The whole 5 by 6 rectangle can be measured by adding the areas of its parts, and
the target region is one of them: 2 + 3 + 2 + 1 12 + 1 + 3 + A = 5 x 6 = 30, or
12 12 +A = 30.
14. B
15. C
24.2 Volume
1. a. meters (yards okay)
b. L or dm3 (or gallons, cu. ft.)
c. cm2 or dm2 (or sq. ins. or possibly sq. ft.)
d. degrees (or cm)
2. 7000
3. a. 6 dm2
b. 80 m2
c. 14 km3
d. 102 cm3
3'. a. 6 dm2
b. 80 m2
c. 45 dm3
d. 50 dm3
4. 1000
5. 8:27
6. a. 5000
b. 0.0027
Shapes and Measurement Page 92
3/9/16
c. 1 000 000
7. Use meter sticks for a framework to show that there would be 100 x 100 cm3 in a
bottom layer of a m3, and that it would take 100 such layers to make the whole
cubic meter.
8. a. 2700
b. 1530
c. 1.2
d. 2.153
e. 0.023
f. 4000
g. 2.257
i. 232 000
h. 400 000
j. 300
9. Area is a measurement of how much it takes to cover a 2D region or a 3D surface;
volume is a measurement of how much it takes to fill or match a 3D container or
shape. 2D regions--for example, rectangular regions--have area without volume.
10. Surface area is a measurement of how much it takes to cover a 3D region; volume
is a measurement of how much it takes to fill or match a 3D container or 3D
region. For example, a 2 cm by 2 cm by 2 cm cube takes 8 cubic centimeters to
match, but 24 square centimeters to cover.
11. First shape: Surface area = 46 square regions
Second shape: S.A. = 46 square regions
Volume = 19 cubes
V = 17 cubes
12. D
25 Counting Units Fast: Measurement Formulas
1. 31.8 cm
1'. A
2. 14 cm2
3. 80 square units
4. 24 m2
4'. D
5. Expected: An argument based on layers; each 1-unit-high layer has volume
numerically equal to the area. The height tells how many such layers there are.
6. V  13 Bh  13 ( 2.5 2 )7  45.8 cm3
Shapes and Measurement Page 93
3/9/16
6'. B
7. A  n(a  c)  12 nc, or algebraic equivalents like na  nc2
7'. A = 112 cm2
7". D
8. Expected: Rotation of 180˚ at the midpoint of one of the non-parallel sides, to give a
parallelogram with base (Base1 + Base2). The original trapezoid will have half
that area.
9. Expected: Rotate 180˚ at the midpoint of x or z to get a parallelogram with base y
and height p. A = 12 pm here.
10. a. In the sketch, the square should have sides 3 cm, with the other sides of the
triangles 4.5 cm.
b. About 11.8 cm3
11. D (Note the expectation is that the type of cone is indeed the mathematical one.)
12. 20 cm
13. 6 cm2
14. 21 cm2
15. 5 14 , 11 14 (from 1 12  3 12 and 2 12  4 12 )
16. S.A. = 36π, or 113.1 cm2
Volume = 36π, or 113.1 cm3 (Instructor: Follow up: “Will this numerical equality
of surface area and volume always happen?”)
17. a. Sketch should show a (b) right circular cylinder.
c. Volume = 96π, or about 301, cm3
Surface area = 80π, or about 251, cm2
18. Mark according to your expectation from class work.
19. 9 as base: h = 5 13 units (altitude drawn outside triangle)
6 as base: h = 8 units (altitude drawn outside triangle)
13 as base: h =
48
13
 3 139 units
(altitude drawn inside triangle)
20. A
21. A
Shapes and Measurement Page 94
3/9/16
22. D
23. C
24. D
25. Area = 70  12  4 2  12  32  70  12.5  109.3 cm2
Perimeter = 24  3  4  46 cm
26. (Each angle of the regular pentagon is 108˚, so the angle of the sector is 360 – 108
= 252 degrees.)
Area =
252
360
  42 
Perimeter =
252
360
56
5
  35.2 cm2
 2 4  32 
28
5
  32  49.6 cm
27. Area = 12  32  12  5 2  17  53.4 cm2
Perimeter = 3  5  3 5  2  8 10  35.1 cm
26.1 Pythagorean Theorem
233  15.3 cm
1.
2. (21 – 15) m x $7/m = $42
2'. A
m 2  n 2 cm
3.
4. Look for work. perimeter = 36 cm
area = 60 cm2
5. a. 72 cm
b, 264 cm2
c. 108 cm
d. 1.52  270  607.5 cm3
6. A
7. P = 6π + 20, or about 38.8, cm
A = 18π + 48, or about 76.5, cm2
8. 245.5 m2
Shapes and Measurement Page 95
3/9/16
9.
288 , or about 17, cm
10. The square of the length of the hypotenuse will equal x2 + y2.
11. 1152 cm3
12. a. 10 dm
b. One possibility (there are others):
15 dm
10 dm
8 dm 12 dm
10 dm
10 dm
10 dm
10 dm
15 dm
c. Volume = 720 dm3
10 dm
Surface area = 576 dm2
13. A  600  12  20 2  600  200  1228.3 cm2; P  80  20  142.8 cm
14. B
15. B
16. The remaining angle in the triangle has 30˚, so the arc of the sector has 330˚. The
Pythagorean theorem gives 16 cm for the length of the hypotenuse, from
13.92  82  h2 .
2
1
2
Area = 330
360  3  2  8 13.9  81.5 cm
Perimeter = 10.9  8  (16  3)  330
360 2 3  49.2 cm
26.2 Some Other Kinds of Measurements
1. a. Briefly, what direct measurement is, use of standard units, measurements (except
for counts) are approximate, measure a whole by measuring its parts and adding.
b. The fourth key idea does not apply to rates. For example, the rate for a trip of 500
miles in 10 hours cannot always be determined by looking at parts—e.g., 120 miles
in 2 hours, plus 380 miles in 8 hours.
Shapes and Measurement Page 96
3/9/16
2. Daniela (67.5 mph). Ann and Ben, 60 mph; Carlo, < 60 mph
3.
279 42
9012
321
 102
 3.147
Shapes and Measurement Page 97
3/9/16