Chapter 1: Foundations for Geometry/Review
Assignment Sheet
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1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Name
Video: Points, Lines Planes and Angles
Understanding Points, Lines and Planes
Measuring Segments and Angles
Video: Constructions 1
Performing Constructions 1
Applying Constructions 1
Video: Constructions 2
Performing Constructions 2
Labeling Relationships
Discovering the Distance Formula (Class Activity)
Calculating Distance and Midpoint
The Hidden Rhombus
Video: Inscribed and Circumscribed Figures
Constructing Inscribed Figures
Geometry in Art (Class Activity)
Chapter Review
Chapter Test
Name ______________________________________________
Completed?
Score _____/10
September/October
14
15
16
17
Pre-Requisite Test
18
Vocabulary
HW: Video for
Sheet #1
21
#2 Understanding
Points, Lines and
Planes
22
#5 Performing
Constructions 1
23
#5 Performing
Constructions 1
#3 Measuring
Segments and
Angles
#6 Applying
Constructions
HW: Video for
Sheet #4
HW: Video for
Sheet #7
28
#9 Labeling
Relationships
29
#10 Discovering
the Distance
Formula (Class
Activity)
24
#6 Applying
Constructions
30
#11 Calculating
Distance and
Midpoint
25
#8 Performing
Constructions 2
1
#12 The Hidden
Rhombus
2
#14 Constructing
Inscribed Figures
HW: Video for
Sheet #13
5
#15 Geometry in
Art (Class
Activity)
6
#16 Chapter
Review
HW: STUDY
7
CHP 1 TEST
8
9
Name __________________________________
CC Geometry
Vocabulary
Point:
Line:
Line Segment:
Ray:
Angle:
Plane:
Shade XYZ :
A
X
Z
Y
R
C
B
Video: Points, Lines, and Planes
Chp 1 Wksht #1
Name __________________________________
CC Geometry
Understanding Points, Lines and Planes
Chp 1 Wksht #2
Name __________________________________
CC Geometry
Measuring Segments and Angles
Chp 1 Wksht #3
Vocabulary:
Congruent: Same size and shape (equal measure)
Midpoint: Separates the segment into two congruent segments
Bisect: To divide into two equal parts
Name each of the following.
A
B
A
C
A
D
A
E
A
F
A
G
A
1.) The point on DA that is 2 units from D
2.) Two points that are 3 units from D
3.) The coordinate of the midpoint of AG
4.) A segment congruent to AC
5.) Use the number line below for a-d. Tell the length of each segment and whether or not the segments are
congruent.
a.) LN and MQ
b.) MP and NQ
c.) MN and PQ
d.) LP and MQ
Hint: When there is no picture, DRAW ONE!
For exercises 10-13, T is the Midpoint of PQ . Find the value of PT for each example.
10.) PT  5x  3 and TQ  7 x  9
11.) PT  4 x  6 and TQ  6 x  2
12.) PT  7 x  24 and PQ  13x  26
13.) XYZ  67 o and XYR  41o find m RYZ .
X
R
Y
Z
14.) XYZ  72 o , XYR  2x  6 , RYZ  4x  12 , find m RYZ .
X
R
Y
Z
15.) YR is the angle bisector of XYZ . If XYR  3x  7 and RYZ  7 x  13 , find m XYZ .
X
R
Y
Z
Name __________________________________
CC Geometry
Video: Constructions 1
Chp 1 Wksht #4
1. Copying a segment
(a) Using your compass, place the pointer at Point
A and extend it until reaches Point B. Your
compass now has the measure of AB.
A
B
(b) Place your pointer at A’, and then create the
arc using your compass. The intersection is the
same radii, thus the same distance as AB. You
have copied the length AB.
Examples
Copy the given segment.
A
B
A'
B
A'
Create the length 3AB
A
2. Bisect a segment
(a) Given AB
(b) Place your pointer at A, extend your compass so that the
distance exceeds half way. Create an arc.
A
(c) Without changing your compass measurement, place your point
at B and create the same arc. The two arcs will intersect. Label
those points C and D.
(d) Place your straightedge on the paper so that it forms CD . The
intersection of CD and AB is the bisector of AB .
B
Examples:
1. Bisect the segment (find the midpoint).
2. Cut the segment into four equal pieces
D
A
B
C
3. Copy an angle
(a) Given an angle and a ray.
(b) Create an arc of any size, such that it intersects both rays of the
angle. Label those points B and C.
(c) Create the same arc by placing your pointer at A’. The
intersection with the ray is B’.
(d) Place your compass at point B and measure the distance from B
to C. Use that distance to make an arc from B’. The intersection of
the two arcs is C’.
(e) Draw the ray A ' C '
A
A'
4. Construct a line parallel to a given line through a point not on the line.
(a) Given a point not on the line.
(b) Place your pointer at point B and measure from B to C. Now
place your pointer at C and use that distance to create an arc. Label
that intersection D.
(c) Using that same distance, place your pointer at point A, and
create an arc as shown.
(d) Now place your pointer at C, and measure the distance from C
to A. Using that distance, place your pointer at D and create an arc
that intersects the one already created. Label that point E.
(e) Create AE .
Question: What type of angles prove these lines are parallel?
A
B
C
Name __________________________________
CC Geometry
COPYING A SEGMENT
1. Given AB, CD, & EF . Use the copy segment
construction to create the new lengths listed below.
Performing Constructions 1
Chp 1 Wksht #5
A
C
E
B
D
F
3AB
CD + EF
2CD + 1AB
EF - CD
CONSTRUCTING A MIDPOINT
2. Given AB & CD . Use the midpoint construction to find the midpoint of AB & CD
D
A
B
C
3. Use your midpoint construction to determine the exact length of
E
1
EF
4
F
4. Given ABC . Make a copy of ABC , A ' B ' C ' .
A
C
B
B'
5. Given DEF . Make a copy of DEF , D ' E ' F ' .
E
D
F
E
E'
PRACTICE - CONSTRUCTION BASICS #1
1. Given MN , construct 2.5 MN
M
2. Given GH , construct 1.75 GH
G
N
H
3. Given ABC, construct a copy of it, A’B’C’.
A
B
C
B'
A
4. Given ABC, can you think of a way to create a line parallel to AB through point C?
(Hint: How could copying an angle help you?)
A
B
C
5. Create a parallel line to DE through point F.
E
D
F
Name __________________________________
CC Geometry
Applying Constructions 1
Chp 1 Wksht #6
1. Convert the mathematical symbols to words.
a)
AB
_______________________
b)
AB
_______________________
c)
AB
_______________________
d)
AB
_______________________
e)
ABC
_______________________
f)
mABC
_______________________
2. Which geometric instrument would I use to measure the length of a segment, the compass or the straightedge?
Explain your answer.
3. What is the difference between drawing and constructing something? So for example, what is the difference
between drawing a perpendicular line and constructing a perpendicular line?
4. A student has done the following construction. What was this student attempting to construct? Is there more than
one thing that the student could be constructing? Explain.
C
A
B
D
5. After learning the midpoint construction, Sally realizes that she could determine one-fourth the length of a
segment. How could she do this? Explain & Diagram.
6. When given AB & CD , a student uses her compass to measure them and then construct a new length EF. What is
the exact length of EF ?
C
D
E
A
B
F
7. A teacher instructs the class to construct the midpoint of a segment. Jeff pulls out his ruler and measures the
segment to the nearest millimeter and then divides the length by two to find the exact middle of the segment. Has he
done this correctly?
8. What is the difference between CD and CD ?
9. Given three possible correct names for the given angle.
A
1
C
B
10. When do we use = and when do we use  ?
11. What does it mean to bisect something?
12. A rhombus is a quadrilateral with 4 congruent sides. Hidden in this construction is a rhombus, can you find it and
then explain why it MUST be a rhombus.
C
A
B
D
Name __________________________________
CC Geometry
Video: Constructions 2
Chp 1 Wksht #7
1. Construct the perpendicular bisector of a line segment
(a) Given AB
A
(b) Place your pointer at A, extend your compass so that
the distance exceeds half way. Create an arc.
B
(c) Without changing your compass measurement, place
your point at B and create the same arc. The two arcs
will intersect. Label those points C and D.
(d) Place your straightedge on the paper and create CD .
2. Construct a line perpendicular to a given segment through a point on the line.
(a) Given a point on a line.
(b) Place your pointer a point A. Create arcs equal
distant from A on both sides using any distance. Label
the intersection points B and C.
(c) Place your pointer on point B and extend it past A.
Create an arc above and below point A.
(d) Place your pointer on point C and using the same
distance, create an arc above and below A. Label the
intersections as points D and E.
(e) Create DE .
Example: Construct the perpendicular line through a point on the line.
A
3. Construct a line perpendicular to a given line through a point not on the line.
(a) Given a point A not on a line.
(b) Place your pointer a point A. Create arcs equal
distant from A on both sides using any distance. Label
the intersection points B and C.
A
(c) Place your pointer on point B and extend it past A.
Create an arc below point A.
(d) Place your pointer on point C and using the same
distance, create an arc below A. Label the intersections
as points D.
(e) Create DE .
Example: Construct the perpendicular line through a point not on the line.
4. Bisect an angle
(a) Given an angle.
(b) Create an arc of any size, such that it intersects both
rays of the angle. Label those points B and C.
(c) Leaving the compass the same measurement, place
your pointer on point B and create an arc in the interior
of the angle.
(d) Do the same as step (c) but placing your pointer at
point C. Label the intersection D.
(e) Create AD . AD is the angle bisector.
Example: Bisect the angle
A
Name __________________________________
CC Geometry
Performing Constructions 2
Chp 1 Wksht #8
Constructing the Perpendicular Bisector (a  line through the midpoint of a segment).
1. Given AB . Use the midpoint construction to construct the perpendicular bisector.
A
A
B
B
Construct the perpendicular line THROUGH A POINT ON THE LINE.
2. Work backwards from the midpoint construction.
A
C
Construct the perpendicular line THROUGH A POINT not on THE LINE.
3. Work backwards through the midpoint construction.
B
A
Construct the angle bisector.
4. Given A, construct the angle bisector, ray AD .
A
A
5. Given sides of a rectangle. Construct the rectangle.
Hint - We need perpendicular lines through A and through M.
A
B
6. Given the side of a square. Construct the square.
D
C
Name __________________________________
CC Geometry
Labeling Relationships
Chp 1 Wksht #9
1. Use the diagram to complete the relationship.
A
C
A
D
A
M
B
o
o
B
B
C
a) ________= ________
c) ________= ________
e) ________= ________
g) ________= ________
b) ________ ________
d) ________ ________
f) ________ ________
h) ________ ________
2. Choose which construction matches the diagram.
A
C
C
D
A
B
C
B
A
C'
B
A'
C
A
B
B'
D
a) The Midpoint of BC
b)  line through A
c)  bisector
d) Copy a segment
a) Copy 
b)  bisector
c)  bisector
d) Copy a segment
a) Copy 
b)  bisector
c)  bisector
d) Copy a segment
a) The Midpoint of BC
b)  line through A
c)  bisector
d) Copy a segment
3. A rhombus is a quadrilateral with 4 congruent sides. Hidden in this construction is a rhombus, can you find it and
then explain why it MUST be a rhombus.
C
D
A
B
4. If you are told that MN is the perpendicular bisector of BC where point M is on BC . Draw the diagram and
completely label it with all known relationships.
5. If you are constructing the perpendicular line through point A (A is on the line), determine the next step.
Step #1 – Place compass at point A, and create two intersections B & C on either side of point A.
Step #2 – Place compass pointer at point B and extend its measure beyond A and make an arc above
and below point A.
Step #3 -- __________________________________________________________________________
Construct a perpendicular line through point B. On the right side, list all the steps of the construction
and label any relationships on the diagram.
B
Construct the midpoint of segment AB. On the right side, list all the steps of the construction and label
any relationships on the diagram.
A
B
Construct the angle bisector of angle A. On the right side, list all the steps of the construction and label
any relationships on the diagram.
A
Name __________________________________
CC Geometry
Discovering the Distance Formula
Chp 1 Wksht #10
1. Consider the picture of two points, C(6, 7) and D(9, 11). We are interested in the distance between them.
a) Draw in line
segment CD using a
straightedge.
y
D
11
c) Draw in the 2 legs
of right triangle
CDE .
7
C
b) Draw in and label
point E, which is
located 3 units right
of point C, and 4
units down from
point D.
d) Label CE with its
length.
e) Label DE with its f) Label point D with
its coordinates
length.
(9,11) .
g) Using the most important theorem in all
of Geometry, use the right triangle to
determine the length of the hypotenuse.
6
j) Our goal now is to
see how to combine
the coordinates of
endpoint C (6,7 )
with endpoint
D (9,11) to come up
k) Write down only
the x-coordinates of
points C and D:
Point C x-coordinate:
with the distance of
5.
9
x
l) Notice that the
horizontal leg CE is
3 units long: How
could you combine
the two numbers
from (k) to get a
result of 3 on your
calculator?
h) So, what is the
distance between
points C and D?
i) Label point E with
its coordinates (9, 7)
m) Write a
mathematical
expression to
represent the idea
from (l). Your
expression should
have the numerals 9
and 6 in it:
n) Write down only
the y-coordinates of
point C and D:
Point D x-coordinate:
.
Point C y-coordinate:
Point D y-coordinate:
(Put parentheses
around it.)
o) Notice that the
p) Write a
mathematical
expression to
represent the idea
from (o). Your
expression should
have the numerals
11 and 7 in it:
vertical leg DE is 4
units long: How
could you combine
the two numbers
from (n) to get a
result of 4 on your
calculator?
q) Looking at Pythagorean Theorem work from part (g), we see the
basic equation:
32  42  52 :
Rewrite this equation by replacing the 3 with (9  6) (don't simplify
it!), and replace the 4 with (11  7) (again, don't simplify).
(Put parentheses
around it.)
r) Let's relabel the numbers for points C and
D. Instead of numerals, let's use symbols.
We'll call point C "Point 1," and label it
accordingly:
 
C
(
6
 
,
7
s) Now, let's rewrite your answer from (q):
- Replace the 6, 7, 9, and 11 with x1 , y1 , x2 , and y2 (Make sure you
match them up as specified in part (r);
- Replace the 5 with a "d" to represent the distance:
)
And we'll label point D as "Point 2":
 
D
(
9
 
,
11
)
2. You should now be looking at the equation ( x2  x1 ) 2  ( y 2  y1 ) 2  d 2 :
a) Rewrite it so the d 2 term is on the
left side, and the radical part is on
the right side:
b) Take the square root of both
sides:
c) Simplify. Your equation should
begin:
d=
3. Let's summarize: the distance formula for the distance d between two points ( x1 , y1 ) and ( x2 , y2 ) is
d  ( x2  x1 ) 2  ( y 2  y1 ) 2
a) Let's apply the formula to figure out the distance
between ( 2, 8) and (7, 20) :
b) Copy the formula:
First, label the points ( x1 , y1 ) and ( x2 , y2 ) : (Put the little
labels over them):
( 2, 8) and (7, 20)
c) Substitute:
d) Simplify. Answer in simplified radical form if necessary.
4. Now, let's apply the formula to figure out the distance between the points (4, –6) and (9, –14):
a) Label the points by putting the little labels over them:
b) Copy the formula:
(4, –6) and (9, 14)
c) Substitute:
d) This problem is slightly different from before: What's
different this time?
e) Simplify, being very careful with the negatives:
f) Simplify, answering in simplified radical form if
necessary:
Name __________________________________
NR Geometry
Calculating Distance and Midpoint
Chp 1 Wksht #11
Formulas:
Midpoint
 x 2  x1 y 2  y1 
,


2 
 2
Distance
d  ( y 2  y1 ) 2  ( x 2  x1 ) 2
Use the distance formula to determine the distance between each pair of points. Answer in simplified radical form
when appropriate.
1) (3, 10) and (–4, 31)
4.)
6.)
8.)
2) (–2, –12) and (4, –9)
5.)
7.)
9.)
3) A(3, 4) and B(-2, 4)
Name __________________________________
NR Geometry
The Hidden Rhombus
Chp 1 Wksht #12
Why do these constructions work?
Construct a Midpoint
Construct a Perpendicular Line
Through a Point Not on the Line
Construct a Perpendicular Line Through a
Point On the Line
Construct an Angle Bisector
1. When performing these constructions you kept your compass at a
fixed length. This creates distances that are equal - thus the 4 equal
sides of a rhombus.
a) Use a ruler and draw in the missing sides of the rhombus in each of
these constructions.
b) Shade the rhombus with your pencil.
2. In the first three constructions we needed to create lines that were perpendicular.
Which property of a rhombus do you think we are using to accomplish this goal?
Opposite sides are ||
Opposite sides are 
Opposite angles are 
Consecutive ’s = 180
Diagonals bisect each other
Diagonals are 
Diagonals are  Bisectors
3. In the angle bisector construct we need to divide an angle into two equal parts.
Which property of a rhombus do you think we are using to accomplish this goal?
Opposite sides are ||
Opposite sides are 
Opposite angles are 
Consecutive ’s = 180
Diagonals bisect each other
Diagonals are 
Diagonals are  Bisectors
4. Given VB -- perform the midpoint construction. This time labeling the two intersection found to be H and K. Draw in
VH , VK , BH , & BK . Also draw HK .
V
B
Why is VH = VK? _______________________________________________________________________
Why is BH = BK? _______________________________________________________________________
Why is VH = VK = BH = BK? _______________________________________________________________
What is the most specific name for the quadrilateral VHBK? _____________________________________
Will this specific quadrilateral be formed every time using this construction? Yes or No
Why or why not…
Label the intersection of HK and VB is point M.
What is true about VM and BM?
_______________________________________________________
What is true about HM and KM?
_______________________________________________________
What is the measure of the angle formed at the intersection of HK and VB ? _________________
Name __________________________________
NR Geometry
Video: Inscribed and Circumscribed Figures
Chp 1 Wksht #13
1. The construction of an inscribed equilateral.
(A) Given Circle A
(B) Create a diameter BC
(C) Create a circle at C with radius AC . Label the two
intersections D and E.
A
(D) Create BD , BE & ED
Example:
Construct an inscribed Equilateral, label all congruent
parts.
2. The construction of an inscribed square.
(A) Given Circle A
(B) Create a diameter BC
(C) Construct a perpendicular line to BC through A.
(D) Create BD , DC , CE & EB
Example:
Construct an inscribed
Square.
A
3. The construction of an inscribed hexagon.
(A) Given Circle A
(B) Create a diameter BC
A
(C) Create a circle at C with radius AC . Label the two
intersections D and E.
(D) Create a circle at B with radius BA . Label the two
intersections F and G.
(E) Create CD , DF , FB , BG , GE & EC
Example:
Construct an inscribed Hexagon.
Inscribed vs. Circumscribed
Inscribed:
Circumscribed
Name __________________________________
NR Geometry
Constructing Inscribed Figures
Chp 1 Wksht #14
1. Determine whether the relationship is INSCRIBED or CIRCUMSCRIBED.
a) The triangle is _____________.
b) The hexagon is _____________
c) The circle is _______________
d) The hexagon is _____________
e) The circle is _______________
f) The triangle is ______________
2. Jeff uses his compass to make a cool design. He just keeps creating congruent circles… over and over…
a) Find a regular hexagon (shade it in)
b) Find a different regular hexagon (shade it in)
c) Find an equilateral triangle (shade it in)
d) Find a different equilateral triangle (shade it in)
3. The inscribed equilateral triangle has a central angle of 120 because 360 / 3 = 120, an inscribed square has a
central angle of 90 because 360 / 4 = 90. The central angle of a decagon is 36 because 360 / 10 = 36. Use this
information and a compass to create an inscribed decagon.
36°
4. Construct the requested inscribed polygons.
a) Construct an equilateral triangle inscribed in the
provided circle using your compass and straightedge.
b) Construct a square inscribed in the provided circle using
your compass and straightedge.
5. Construct the requested inscribed polygons.
a) Construct a regular hexagon inscribed in the provided
circle using your compass and straightedge.
b) Construct a regular octagon inscribed in the provided
circle using your compass and straightedge.
Hint: The central angle is 45, half of
the square’s central angle of 90.
Name __________________________________
NR Geometry
Geometry in Art
Chp 1 Wksht #15
Now that you know the basics of constructions you have the ability to create some very cool
Geometric Art. Using just the construction of the inscribed hexagon (and the addition of a few
segments and arcs) you can create a number of very interesting designs. Here are a few examples
of Geometric Art. (The QR code links to an introduction video.)
Practice Area (Shading & coloring change the look of the shape…. Also symmetry is very appealing to the eye.)
Final Copy (Full Color & Clarity)
Name: ____________________ Period: ____
Name __________________________________
NR Geometry
Chapter 1 Review
Chp 1 Wksht #16
Know the following definitions and any appropriate symbols (ie segment AB = AB )
Point
Plane
Line
Line Segment
Ray
Angle
Congruent
Perpendicular
Supplementary
Complementary
Bisect
Bisect
Collinear
Noncollinear
Coplanar
Noncoplanar
Adjacent
Vertical Angles
Midpoint
Distance
Acute
Obtuse
Straight Angle
Right Angle
Area
Supplement
Complement
Review the basic constructions.
1. Copying Segments
A
B C
D
a) 4AB – CD
b) 2.75CD + AB
2. Midpoint Construction
a) Given diameter AB, find the center
b) Which is larger AB or ½ CD? ____________
B
B
C
A
D
A
Copy them here to compare them
c) Construct the midpoint of AB
d) CD 
1
GH . Construct GH .
3
A
D
C
B
G
3. Copying an Angle & Parallel Line Construction
a) Create A’B’C’ by copying ABC.
b) Create D’E’F’ by copying DEF.
D
E
A
F
C
B
E'
B'
c) Create A’ by copying Copy A
d) Create E’ such that mE’ = 2(mE)
E
A
A'
E'
4. Angle Bisector Construction
a) Construct the angle bisector of A.
b) Create DBC such that mDBC = ½ (mABC)
A
B
C
A
5. Perpendicular Constructions
Construct the following.
a) the perpendicular bisector of AD
b) a perpendicular line to AE through E
c) the perpendicular line to DA through C
C
D
A
E
6. Inscribed Polygons
a) Inscribed Square
b) Inscribed Equilateral Triangle
c) Inscribed Hexagon
7. Labeling and Communicating Relationships.
a.) Perform the perpendicular bisector construction. Label all relationships on the diagram and list the steps on
the right.
B
A
b.) Julia is constructing a perpendicular through a point on the line. What is the next step? Be very specific as to
what he should do next.
C
A
B
1. Given a point on a line.
2. Place your pointer a point A. Create arcs equal
distant from A on both sides using any distance.
Label the intersection points B and C.
3. Place your pointer on point B and extend it past
A. Create an arc above and below point A.
4.