PRACTICE FINAL
Multiple Choice
Identify the choice that best completes the statement or answers the question.
Refer to Figure 1.
m
n
B
A
H
D
C
K
J
G
F p
Figure 1
____
1. What is another name for line n?
a. line JB
c.
b.
____
____
____
d. AC
2. Name the intersection of lines m and n.
a. K
b.
c. B
d. D
3. What is another name for line m?
a. line JG
b.
c. DB
d. line JB
4. Name a line that contains point A.
a.
c.
b. m
____
____
5. Name a point NOT contained in
a. K
b. A
or
.
c. H
d. D
6. Which of these is NOT a way to refer to line BD?
a.
c.
b. m
____
K
d. DB
7. Name three points that are collinear.
d. line JD
B
A
H
D
G
J
C
F
a. B, G, F
b. C, D, H
____
c. J, G, F
d. J, D, G
8. Name three points that are collinear.
K
R
S
L
T
P
M
N
Q
a. M, L, R
b. L, P, T
____
c. Q, L, M
d. R, S, K
9. Are points A, C, D, and F coplanar? Explain.
D
H
G
F
A
P
C
B
a. Yes; they all lie on plane P.
b. No; they are not on the same line.
c. Yes; they all lie on the same face of the pyramid.
d. No; three lie on the same face of the pyramid and the fourth does not.
Refer to Figure 2.
B
L
A
D
K
C
F
G
Figure 2
____ 10. How many planes contain points B, C, and A?
a. 1
c. 0
b. 2
d. 3
____ 11. Name an intersection of plane GFL and the plane that contains points A and C.
a. line LC
c. line AC
b. C
d. plane CAB
Find the measurement of the segment.
____ 12.
mm,
P
mm
R
S
PS = ?
a. 32.7 mm
b. 5.1 mm
c. 32.5 mm
d. 32.4 mm
Use the number line to find the measure.
G
–8
–7
____ 13. RK
a. 2
b. 5
H
–6
–5
K L
–4
–3
–2
M
–1
0
1
N P Q
2
3
4
5
R
6
7
8
c. 7
d. 10
Find the coordinates of the midpoint of a segment having the given endpoints.
____ 14.
a.
c.
b.
d.
Measure the angle.
____ 15.
a. 40
b. 140
c. 29
d. 151
In the figure,
bisects
.
F
x
G
y
K
H
____ 16. If
a. 7
b. 14.43
and
In the figure,
and
, find w.
c. 52
d. 3.5
are opposite rays.
J
P
1
K
2
3 4
N
M
L
and
bisects
.
____ 17. What bisects
a. P
?
c.
b.
d.
____ 18. Which is NOT true about
?
a.
is acute.
b.
c. Point M lies in the interior of
d. It is an angle bisector.
____ 19. If
a. 153
b. 33
and
____ 20. If
a. 137
b. 12
and
____ 21. If
a. 5.57
b. 41
and
.
, what is
, what is
?
c. 27
d. 12
?
c. 4.2
d. 43
, what is
?
c. 12
d. 21.71
Use the figure to find the angles.
I
H
G
J
2
1
M
Q
K
L
____ 22. Name two acute vertical angles.
a.
b.
c.
d.
____ 23. Name a linear pair.
a.
b.
c.
d.
____ 24. Name an angle supplementary to
a.
b.
____ 25. Name two obtuse vertical angles.
.
c.
d.
a.
b.
c.
d.
____ 26. The measures of two complementary angles are
and
a. 42, 48
c. 8.75
b. 4.25
d. 96, 84
____ 27. The measures of two complementary angles are
and
a. 117, 63
c. 63, 27
b. 19
d. 10
. Find the measures of the angles.
. Find the measures of the angles.
Name each polygon by its number of sides.
____ 28.
a. hexagon
b. triangle
c. pentagon
d. quadrilateral
a. dodecagon
b. pentagon
c. hexagon
d. decagon
____ 29.
Name each polygon by its number of sides. Then classify it as convex or concave and regular or irregular.
____ 30.
a. pentagon, convex, regular
b. hexagon, concave, regular
c. hexagon, convex, regular
d. hexagon, convex, irregular
____ 31.
a. quadrilateral, convex, regular
b. quadrilateral, concave, irregular
c. pentagon, convex, irregular
d. quadrilateral, convex, irregular
a. quadrilateral, convex, regular
b. quadrilateral, concave, irregular
c. pentagon, convex, regular
d. quadrilateral, convex, irregular
____ 32.
Find the length of each side of the polygon for the given perimeter.
____ 33.
units. Find the length of each side.
2n + 2
2n – 3
a.
b.
c.
d.
____ 34.
18 units, 18 units, 13 units
18.66 units, 18.66 units, 13.66 units
27 units, 27 units, 22 units
26 units, 26 units, 21 units
units. Find the length of each side.
2t + 2
2t
2t + 4
a. 16 units, 18 units, 14 units
b. 12 units, 16 units, 8 units
c. 10 units, 18 units, 8 units
d. 12 units, 14 units, 10 units
Find the circumference of the figure.
____ 35.
3.8 in.
a. about
b. about
c. about
d. about
Find the area of the figure.
____ 36.
3.1cm
2.3 cm
a. 7.13
b. 5.4
c. 71.3
d. 10.8
____ 37.
6m
a. 36
b. 18
c. 12
d. 72
Identify the solid.
____ 38.
F
a. cone
b. sphere
c. cylinder
d. prism
____ 39. Name the bases of the solid.
A
B
C
a.
b.
and
and
Name the edges of the solid.
D
c.
d.
and
and
____ 40.
B
C
D
A
G
F
E
a.
b.
c.
d.
H
,
,
,
,
,
,
,
,
,
,
, and
, and
, and
,
,
,
,
,
,
, and
Name the faces of the solid.
V
____ 41.
U
Z
X
W
a.
b.
c.
d.
Y
WUY, XVZ, UVZY, UVXW, and WXZY
WUY, XVZ, UVZY, and UVXW
UVZY, UVXW, and WXZY
WUY and XVZ
Make a conjecture about the next item in the sequence.
____ 42.
a. 1024
b. 1025
c. 4096
d. 1022
Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.
____ 43. Given: a concave polygon
Conjecture: It can be regular or irregular.
a. False; to be concave the angles cannot be congruent.
b. True
c. False; all concave polygons are regular.
d. False; a concave polygon has an odd number of sides.
____ 44. Given: Point B is in the interior of
Conjecture:
a. False;
may be obtuse.
b. True
.
c. False; just because it is in the interior does not mean it is on the bisecting line.
d. False;
.
____ 45. Given:
Conjecture:
a. False;
b. True
c. False;
d. False;
____ 46. Given: points R, S, and T
Conjecture: R, S, and T are coplanar.
a. False; the points do not have to be in a straight line.
b. True
c. False; the points to not have to form right angles.
d. False; one point may not be between the other two.
____ 47. Given: Two angles are supplementary.
Conjecture: They are both acute angles.
a. False; either both are right or they are adjacent.
b. True
c. False; either both are right or one is obtuse.
d. False; they must be vertical angles.
B
____ 48. Given:
A
C
Conjecture:
a. False; the angles are not vertical.
b. True
c. False; the angles are not complementary.
d. False; there is no indication of the measures of the angles.
____ 49. Given: segments RT and ST; twice the measure of
Conjecture: S is the midpoint of segment RT.
a. True
b. False; point S may not be on .
c. False; lines do not have midpoints.
d. False;
could be the segment bisector of .
.
Use the following statements to write a compound statement for the conjunction or disjunction. Then find its
truth value.
p: An isosceles triangle has two congruent sides.
q: A right angle measures 90
r: Four points are always coplanar.
s: A decagon has 12 sides.
____ 50. p and r
a. An isosceles triangle has two congruent sides and four points are always coplanar; false.
b. An isosceles triangle has two congruent sides and a decagon has 12 sides; false.
c. An isosceles triangle has two congruent sides and four points are always coplanar; true.
d. An isosceles triangle has two congruent sides and a decagon has 12 sides; true.
____ 51.
a. Four points are always coplanar, or a right angle measures 90 and a decagon has 12 sides;
false.
b. Four points are always coplanar, and a right angle measures 90 or a decagon has 12 sides;
true.
c. Four points are always coplanar, or a right angle measures 90 and a decagon has 12 sides;
true.
d. Four points are always coplanar, and a right angle measures 90 or a decagon has 12 sides;
false.
____ 52.
a. A right angle measures 90 or four points are always coplanar, and a decagon has 12 sides;
false.
b. A right angle measures 90 and four points are always coplanar, or a decagon has 12 sides;
false.
c. A right angle measures 90 and four points are always coplanar, or a decagon has 12 sides;
true.
d. A right angle measures 90 or four points are always coplanar, and a decagon has 12 sides;
true.
____ 53. s or q
a. A decagon has 12 sides or a right angle measures 90; true.
b. A decagon has 12 sides or a right angle measures 90; false.
c. A decagon has 12 sides and a right angle measures 90; true.
d. A decagon has 12 sides and a right angle measures 90; false.
____ 54.
a.
b.
c.
d.
An isosceles triangle has two congruent sides and a decagon has 12 sides; true.
An isosceles triangle has two congruent sides or a decagon has 12 sides; false.
An isosceles triangle has two congruent sides or a decagon has 12 sides; true.
An isosceles triangle has two congruent sides and a decagon has 12 sides; false.
a.
b.
c.
d.
Four points are always coplanar and a right angle measures 90; false.
Four points are always coplanar and a right angle measures 90; true.
Four points are always coplanar or a right angle measures 90; false.
Four points are always coplanar or a right angle measures 90; true.
____ 55.
Complete the truth table.
____ 56.
a.
b.
c.
d.
Write the inverse of the conditional statement. Determine whether the inverse is true or false. If it is false, find
a counterexample.
____ 57. An equilateral triangle has three congruent sides.
a. A non-equilateral triangle has three congruent sides. False; an isosceles triangle has two
congruent sides.
b. A figure that has three non-congruent sides is not an equilateral triangle. True
c. A non-equilateral triangle does not have three congruent sides. True
d. A figure with three congruent sides is an equilateral triangle. True
____ 58. All country names are capitalized words.
a. All capitalized words are country names. False; the first word in the sentence is
capitalized.
b. All non-capitalized words are not country names. True
c. All non-country names are capitalized words. False; most of the words in the sentence are
non-capitalized words.
d. All non-country names are non-capitalized words. False; the first word in the sentence is
capitalized.
____ 59. Independence Day in the United States is July 4.
a. July 4 is not Independence Day in the United States. False; it is Independence Day.
b. Non-Independence Day in the United States is not July 4. True
c. Non-Independence Day in the United States is July 4. False; July 4 is Independence Day
in the United States.
d. Non-July 4 is not Independence Day in the United States. True
Write the contrapositive of the conditional statement. Determine whether the contrapositive is true or false. If
it is false, find a counterexample.
____ 60. If you are 16 years old, then you are a teenager.
a. If you are not a teenager, then you are not 16 years old. True
b. If you are not 16 years old, then you are not a teenager. False; you could be 17 years old.
c. If you are not a teenager, then you are 16 years old. True
d. If you are a teenager, then you are 16 years old. False; you could be 17 years old.
____ 61. A converse statement is formed by exchanging the hypothesis and conclusion of the conditional.
a. A non-converse statement is not formed by exchanging the hypothesis and conclusion of
the conditional. True
b. A statement not formed by exchanging the hypothesis and conclusion of the conditional is
a converse statement. False; an inverse statement is not formed by exchanging the
hypothesis and conclusion of the conditional.
c. A non-converse statement is formed by exchanging the hypothesis and conclusion of the
conditional. False; an inverse statement is formed by negating both the hypothesis and
conclusion of the conditional.
d. A statement not formed by exchanging the hypothesis and conclusion of the conditional is
not a converse statement. True
____ 62. Two angles measuring 180 are supplementary.
a. Two angles not measuring 180 are supplementary. True
b. More than two angles measuring 180 are non-supplementary. True
c. Non-supplementary angles are not two angles measuring 180. True
d. Non-supplementary angles are two angles measuring 180. False; supplementary angles
must measure 180.
In the figure below, points A, B, C, and F lie on plane P. State the postulate that can be used to show each
statement is true.
D
F
A
P
C
B
____ 63. A and B are collinear.
a. If two points lie in a plane¸ then the entire line containing those points lies in that plane.
b. Through any two points there is exactly one line.
c. If two lines intersect¸ then their intersection is exactly one point.
d. A line contains at least two points.
____ 64. Line AD contains points A and D.
a. If two lines intersect¸ then their intersection is exactly one point.
b. If two points lie in a plane¸ then the entire line containing those points lies in that plane.
c. A line contains at least two points.
d. Through any two points¸ there is exactly one line.
Refer to the figure below.
B
C
A
D
G
F
H
I
____ 65. Name all planes intersecting plane CDI.
a. ABC, CBG, ADI, FGH
b. CBA, DAF, HGF
c. BAD, GFI, CBG, GFA
d. DAB, CBG, FAD
____ 66. Name all segments parallel to
a.
b.
c.
d.
.
____ 67. Name all planes intersecting plane
a.
b.
____ 68. Name all segments skew to
a.
b.
____ 69. Name all segments skew to
a.
b.
.
c.
d.
.
c.
d.
.
____ 70. Name all planes intersecting plane
a.
b.
c.
d.
.
c.
d.
____ 71. Name all segments parallel to
a.
b.
.
c.
d.
Identify the sets of lines to which the given line is a transversal.
____ 72. line j
m
n
o
j
i
a.
b.
c.
d.
p
lines m and n¸ n and o¸ m and o
lines m and p¸ n and o
lines i
lines m and n¸ n and o¸ m and o¸ m and p¸ n and p¸ o and p
____ 73. line a
a.
b.
c.
d.
lines c and b¸ f and d¸ c and f¸ c and d¸ b and d
lines a and b¸ a and c¸ a and d¸ a and f
lines f and d¸ c and f¸ c and d¸ b and d
lines c and b¸ f and d
____ 74. In the figure,
. Find
.
v
125°
q
119°
S
1
p
t
a.
b.
Determine whether
c.
d.
and
are parallel, perpendicular, or neither.
____ 75.
a. perpendicular
b. parallel
c. neither
Write an equation in slope-intercept form of the line having the given slope and y-intercept.
____ 76.
a.
c.
b.
d.
Write an equation in point-slope form of the line having the given slope that contains the given point.
____ 77.
a.
b.
c.
d.
a.
c.
b.
d.
____ 78.
____ 79.
a.
b.
c.
d.
Given the following information, determine which lines, if any, are parallel. State the postulate or theorem
that justifies your answer.
____ 80.
c
O
d
g
Q
H
N
L
f
K
a
M
P
J
b
a.
b.
c.
d.
; congruent corresponding angles
; congruent corresponding angles
; congruent alternate exterior angles
; congruent alternate exterior angles
Construct a line perpendicular to m through P. Then find the distance from P to m.
____ 81. Line m contains points
and
. Point P has coordinates
a.
.
c.
5
5
P
4
–5
–4
–3
–2
3
3
2
2
1
1
–1
–1
1
2
3
4
5
–5
–4
–3
–2
–1
–1
–2
–2
–3
–3
–4
–5
m
P
4
–4
–5
1
2
m
3
4
5
b.
d.
5
5
4
4
3
3
P
2
m
–5
–4
–3
–2
–1
–1
P
2
m
1
1
1
2
3
4
–5
5
–4
–3
–2
–1
–1
–2
–2
–3
–3
–4
–4
–5
–5
1
2
Find the distance between the pair of parallel lines.
____ 82.
a.
b.
c.
d.
Use a protractor to classify the triangle as acute, equiangular, obtuse, or right.
____ 83.
a. right
b. acute
Find the measures of the sides of
c. equiangular
d. obtuse
and classify the triangle by its sides.
____ 84.
a. equilateral
b. obtuse
Find each measure.
____ 85.
c. isosceles
d. scalene
3
4
5
41°
47°
1
2
66°
3
37°
a.
b.
c.
d.
____ 86.
2
55°
3
28°
1
46°
a.
b.
c.
d.
Identify the congruent triangles in the figure.
____ 87.
W
S
R
T
U
V
a.
b.
Determine whether
c.
d.
given the coordinates of the vertices. Explain.
____ 88.
a. Yes; Two sides of triangle PQR and angle PQR are the same measure as the corresponding
sides and angle of triangle STU.
b. Yes; Each side of triangle PQR is the same length as the corresponding side of triangle
STU.
c. No; One of the triangles is obtuse.
d. No; Each side of triangle PQR is not the same length as the corresponding side of triangle
STU.
____ 89. Triangles ABC and AFD are vertical congruent equilateral triangles. Find x and y.
B
C
(2 y+ 6)°
A
2x – 3
x+ 4
D
F
a.
c.
b.
d.
____ 90. Triangle RSU is an equilateral triangle.
bisects
. Find x and y.
R
( y + 12)°
4
x
U
S
T
a.
b.
c.
d.
____ 91. Triangle RSU is an equilateral triangle.
bisects
R
(2 y + 12)°
x
U
4
T
S
a.
c.
b.
d.
Identify the type of congruence transformation.
____ 92.
. Find x and y.
y
x
a. reflection
b. translation
c. rotation
d. not a congruence transformation
Position and label the triangle on the coordinate plane.
____ 93. right isosceles
y
a.
with congruent sides
and
c.
a units long
y
C (0, 1)
O A (0, 0)
C (0, a)
B (1, 0)
y
b.
O A (0, 0)
x
____ 94. isosceles
A (0, a)
B ( a, 0)
with base
x
2b units long
x
y
d.
C (0, a)
O A (0, 0)
B (a, 0)
O
C (a,, a)
B (a, 0)
x
a.
y
c.
M ( b, c)
y
L(0, 2 b)
N ( c, b)
O L (0, 0)
b.
N (2 b, 0)
O M (0, 0)
x
y
d.
L ( b, c)
x
y
N (0, 2 b)
L ( c, b)
O M (0, 0)
____ 95. right
a.
N (2 b, 0)
with hypotenuse
y
O M (0, 0)
x
, leg
x
4a units long, and leg
y
c.
one-fourth the other leg
B (0, 4 a)
C (0, a )
A (a , 0)
O B (0, 0)
b.
A(4 a, 0) x
y
O C (0, 0)
d.
x
y
C (0, 4 a)
B (0, a)
B ( a, 0)
O A (0, 0)
____ 96. equilateral
y
a.
O C (0, 0)
x
with height c units and base
2d units
c.
y
Y (d, c)
O X (0, 0)
Z (2 d, 0)
A(4a, 0) x
Z (c, d)
x
O Y (0, 0)
X (2 d, 0)
x
y
b.
y
d.
Y (0, 2d)
Y (0, 2 d)
Z ( c, d)
Z ( c, d)
O X (0, 0)
____ 97. isosceles
y
a.
O X (0, 0)
x
with
x
half the length of the base and bisecting the base
y
c.
C (a, a)
C (a, a )
A (2 a, 0)
O B (0, 0)
D ( a, 0)
A(2a, 0) x
O B (0, 0) D (a, 0)
y
b.
y
d.
C (a, a )
C (a, a)
(0, 0) B
O B (0, 0)
____ 98.
D (a, 0)
is an altitude,
Z
x
A (2 a, 0)
, and
O
x
. Find
A (2 a, 0)
D (a, 0)
x
.
A
X
W
Y
a. 34
b. 32
C
c. 18
d. 31
Determine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. Explain.
____ 99. 3, 9, 10
a. Yes; the third side is the longest.
b. No; the sum of the lengths of two sides is not greater than the third.
c. No; the first side is not long enough.
d. Yes; the sum of the lengths of any two sides is greater than the third.
____ 100. 9.2, 14.5, 17.1
a.
b.
c.
d.
Yes; the third side is the longest.
No; the first side is not long enough.
Yes; the sum of the lengths of any two sides is greater than the third.
No; the sum of the lengths of two sides is not greater than the third.
Short Answer
101. The graph shows the average daily temperature for different cities. Can you be sure that the temperature of
City H is
more than City G? Explain.
City
J
57.7
I
73.8
H
77.6
G
66.9
F
78.2
0
10
20
30
40
50
60
70
80
Temperature (°F)
102. The longitude-latitude coordinates of Worland, Wyoming are
and of Portland, Maine are
. If Worland is one endpoint of a segment and Portland is its midpoint, find the latitude
and longitude of the other endpoint.
103. A ray of light is reflected when it hits a mirror. The angle at which the light strikes the mirror is the angle of
incidence, i. The angle at which the light is reflected is the angle of reflection, r. The angle of incidence and
the angle of reflection are congruent. In the diagram below, if
, what is the angle of reflection
and
104. In a museum, Nick is looking at a famous painting through a mirror at an angle of
painting makes with the mirror. Also find
and
Find the angle the
105. Keith has made a square that has 6-inch cord sides. He bends the square into a circular loop. What is the
maximum radius of the circular loop? Round to the nearest tenth.
106. Nick framed a square painting that was 30 centimeters long with a decorative strip. He wants to surround a
circular picture frame with the same length of strip. What is the maximum radius of the picture frame? Round
to the nearest tenth.
Angela has made a large pasta stuffed with pumpkin puree. The pasta was in the form of a right triangular
prism. The isometric view of the pasta is shown below.
107. What is the surface area in square feet to the nearest tenth?
Write the contrapositive of the conditional statement. Determine whether the contrapositive is true or false. If
it is false, find a counterexample.
108. Vertical angles are two nonadjacent angles formed by two intersecting lines.
109. Two segments having the same measure are congruent.
Write a paragraph proof.
110.
and
are adjacent angles. Prove that point Q is in the interior of
111. Given: Isosceles triangle GHJ;
Prove:
.
is a segment between points K and L on line KL; HG = JG.
G
H
J
K
L
112. Given:
Prove:
all intersect at point P;
K
and
.
L
J
M
P
O
N
113. Given: Vertical isosceles triangles;
Prove:
Q
bisects
;
.
R
P
S
T
Write a two-column proof.
114. Given: Regular hexagon ABCDFG;
Prove: If
and
, then
A
F
115.
C
D
.
prove
.
B
H
G
each bisect the hexagon and meet at point H.
and
.
. If
,
L
F
D
A
M
N
O
B
H
R
C
P
T
G
116.
are all parallel to each other.
F H
1
A
3
2
G
J
I
L
5
N
6
B
4
K M
and
O
117. Given: Square GHJK
Prove:
G
H
F
K
J
118. Given: W is the midpoint of
Prove:
X
and
;
.
V
W
Z
Y
119. Given: A regular hexagon; M is the midpoint of
Prove:
and
.
. Prove
.
N
R
M
P
O
120. In a group of 150 students, 60 students joined the math club, 54 students joined theater, and 27 students joined
both. Draw a Venn diagram to represent the data.
121. A travel agency surveyed a group of 215 high school students. The survey results showed that 86 students
have been to Mexico, 106 have been to Canada, and 59 have been to both Mexico and Canada. Draw a Venn
diagram to represent the data.
122. Determine the converse of the conditional statement, If a triangle is acute, then it has three acute angles.
State whether the converse is true or false. If the converse is false, find a counterexample.
123. Write an example of a conditional statement with its converse.
Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment. If it does not,
write invalid.
124. (1) If you are a painter, then you are artistic.
(2) If you like colors, then you are artistic.
(3) If you like colors, then you are a painter.
125. Emma did not go swimming as it was raining. If it does not rain, then you can go swimming. Write a
valid conclusion to the hypothesis If it is not raining, . . .
126. Nick marked six noncoplanar and noncollinear points on a sheet of paper. He joined the points to form a
group of connected planes. How many lines and planes will he have?
127. The runways at an airport intersect as shown. The measure of
of
3
4
at the intersection is
Find the measure
128. A carpenter has constructed a chair like the one shown below. Describe the type of lines formed by the front
edge of the seat and a back leg.
In 1995, the circulation of a local newspaper was 1970. In 1997, the circulation was 2330.
129. Predict the circulation for 2001.
The graph below shows the use of laptop computers in a state, in hundred thousands, from 1990 to 1999.
130. If the use of laptop computers continues to increase at the same rate, what will be the number of laptop
computers used in 2006?
John and Nick marked the location of their respective offices on a map. John marked the point
Nick marked the point
131. Write an equation in the slope-intercept form that models the line between the offices of John and Nick.
and
132. Identify the obtuse triangles if
A
and
B
F
D
C
133. Write a two-column proof for the problem.
Given: PQRS is a quadrilateral.
Prove:
P
S
Q
R
Write a flow proof.
134. Given:
Prove:
is isosceles with base
D
F
G
135. Given:
Prove:
H
I
bisects
;
S
R
V
U
.
;
.
136. Write a paragraph proof for the problem.
Given:
Prove:
A
D
B
C
137. Samantha has cut a pastry into four parts. Suppose
Justify your answer.
S
and T is the midpoint of PR. Determine whether
R
T
P
Q
Write a two-column proof.
138. Given:
is equilateral;
Prove: T is the midpoint of
bisects
.
.
R
U
S
T
139. Write a two-column proof for the problem.
Given:
is equilateral;
Prove:
P
Q
R
S
T
.
. Using the given vertices, identify the congruence transformation.
140.
Write a coordinate proof for the statement.
141. If a line is drawn between the midpoints of two sides of an equiangular triangle, then the resulting triangle is
also equiangular.
142. A triangle with coordinates
,
,
is isosceles.
143. If a segment from the vertex of the right angle bisects the hypotenuse of a right isosceles triangle, then the
resulting two triangles are congruent.
144. Triangle ABC has vertices
and
List the angles in order from the greatest to
the least measure.
145. A tree 44 meters high cast a shadow 60 meters long, as shown below. Write an inequality relating x and y.
Then solve the inequality for x in terms of y.
2x°
_y
+ 4°
2
44 m
60 m
Write an indirect proof.
146. Given:
Prove:
is positive.
147. Given:
Prove:
;
A
D
B
148. Given:
C
; A is the midpoint of BF and CD.
Prove:
B
C
A
D
F
149. Given:
Prove:
with angle measures as shown
O
30°
M
110°
40°
N
150. Megan and Sara took part in a 500-meter race on sports day at the school. As the race finished, Megan
claimed that she was the winner of the race. The teacher said that according to the stopwatch, Megan took 10
minutes to complete the race, whereas Sara completed the race in 9 minutes. So, Megan was not the winner of
the race. Explain whether this is an example of indirect reasoning.
PRACTICE FINAL
Answer Section
MULTIPLE CHOICE
1. ANS: B
A line is made up of points with an arrowhead at each end. A, D, and C are points on line n. A line is
represented by ‘line DC’ or
but not just DC.
Feedback
A
B
C
D
Are those points on line n?
Correct!
Are those points on line n?
Is that how a line is named?
PTS: 1
DIF: Average
REF: Lesson 1-1
OBJ: 1-1.1 Identify and model points, lines, and planes.
TOP: Identify and model points, lines, and planes.
2. ANS: D
Two lines intersect in a point. In this case that is point D.
NAT: NCTM GM.2
KEY: Points | Lines | Planes
Feedback
A
B
C
D
What is the intersection of two lines?
Is a line the intersection of two lines?
Is that point on both lines?
Correct!
PTS: 1
DIF: Average
REF: Lesson 1-1
OBJ: 1-1.1 Identify and model points, lines, and planes.
NAT: NCTM GM.2
TOP: Identify and model points, lines, and planes.
KEY: Points | Lines | Planes
3. ANS: D
A line is made up of points and has no thickness or width. It is drawn with an arrowhead at each end. J, D,
and B are points on line m. A line is represented by ‘line JD’ or
but not just JD.
Feedback
A
B
C
D
Are those points on line m?
Is that how you name a line?
Is that how you name a line?
Correct!
PTS:
OBJ:
TOP:
4. ANS:
1
DIF: Average
REF: Lesson 1-1
1-1.1 Identify and model points, lines, and planes.
Identify and model points, lines, and planes.
A
NAT: NCTM GM.2
KEY: Points | Lines | Planes
Line m contains points J, D, and B. Line p contains points G and F. Only line
contains point A.
Feedback
A
B
C
D
Correct!
Is point A on that line?
Is that a line?
What points are on that line?
PTS:
OBJ:
TOP:
5. ANS:
1
DIF: Basic
REF: Lesson 1-1
1-1.1 Identify and model points, lines, and planes.
Identify and model points, lines, and planes.
C
The points not contained in
or
NAT: NCTM GM.2
KEY: Points | Lines | Planes
are J, B, and H. K is the plane.
Feedback
A
B
C
D
Is that a point or the plane?
Is that point on one of the lines listed?
Correct!
Is that point on one of the lines listed?
PTS: 1
DIF: Basic
REF: Lesson 1-1
OBJ: 1-1.1 Identify and model points, lines, and planes.
NAT: NCTM GM.2
TOP: Identify and model points, lines, and planes.
KEY: Points | Lines | Planes
6. ANS: C
The proper way to refer to a line is any 2 points on the line with an arrow above them or “line such-and-such”,
where “such-and-such” is any 2 points on the line. Using three letters is not correct.
Feedback
A
B
C
D
Does line BD contain point J?
Does that line contain points B and D?
Correct!
Are points J and D on line BD?
PTS: 1
DIF: Basic
REF: Lesson 1-1
OBJ: 1-1.1 Identify and model points, lines, and planes.
TOP: Identify and model points, lines, and planes.
7. ANS: A
Collinear points are points on the same line.
NAT: NCTM GM.2
KEY: Points | Lines | Planes
Feedback
A
B
C
D
Correct!
Are those points on the same line?
What does collinear mean?
Are those points on the same line?
PTS: 1
DIF: Average
REF: Lesson 1-1
OBJ: 1-1.2 Identify collinear points.
NAT: NCTM GM.2
TOP: Identify collinear points.
KEY: Collinear Points
8. ANS: A
Collinear points are points on the same line.
Feedback
A
B
C
D
Correct!
Are those points on the same line?
Are those points on the same line?
What does collinear mean?
PTS: 1
DIF: Average
REF: Lesson 1-1
OBJ: 1-1.2 Identify collinear points.
NAT: NCTM GM.2
TOP: Identify collinear points.
KEY: Collinear Points
9. ANS: D
Points that lie on the same plane are said to be coplanar. Three points are always coplanar but if the fourth
point is not on the same plane with the first three, they are not all coplanar.
Feedback
A
B
C
D
Do all four points lie on the same plane? Which plane?
Do all four points lie on the same plane? Which plane?
What does coplanar mean?
Correct!
PTS: 1
DIF: Average
REF: Lesson 1-1
OBJ: 1-1.3 Identify coplanar points.
NAT: NCTM GM.2
TOP: Identify coplanar points.
KEY: Coplanar Points | Intersecting Lines | Lines in Space
10. ANS: A
B, C, and A make up the back face of the prism.
Feedback
A
B
C
D
Correct!
Where is the second plane?
Do the points determine a face of the prism?
Where are the second and third planes?
PTS: 1
DIF: Average
REF: Lesson 1-1
OBJ: 1-1.4 Identify intersecting lines and planes in space.
TOP: Identify intersecting lines and planes in space.
11. ANS: A
The intersection of two planes is a line.
NAT: NCTM ME.1
KEY: Planes | Planes in Space
Feedback
A
B
C
D
Correct!
Can the intersection of two planes be a point?
Is point A on plane GFL?
Can the intersection of two planes be a plane?
PTS: 1
DIF: Average
REF: Lesson 1-1
OBJ: 1-1.4 Identify intersecting lines and planes in space.
TOP: Identify intersecting lines and planes in space.
12. ANS: C
PS has the same length as PR and RS combined.
NAT: NCTM ME.1
KEY: Planes | Planes in Space
Feedback
A
B
C
D
Did you add correctly?
PS contains both PR and RS.
Correct!
Try adding that again.
PTS: 1
DIF: Basic
REF: Lesson 1-2
OBJ: 1-2.1 Measure segments.
NAT: NCTM ME.2 | NCTM ME.2a
TOP: Measure segments.
KEY: Measurement | Line Segments
13. ANS: D
The distance between two points a and b is
or
.
Feedback
A
B
C
D
You are looking for the measure, not the midpoint.
You are looking for the measure, not the half measure.
Add those numbers again.
Correct!
PTS:
OBJ:
NAT:
KEY:
14. ANS:
1
DIF: Average
REF: Lesson 1-3
1-3.1 Find the distance between two points on a number line.
NCTM GM.2 | NCTM GM.2a
TOP: Find the distance between two points on a number line.
Distance | Number Lines | Distance Between Two Points
C
The formula for the midpoint between two points
is
.
Feedback
A
B
C
D
Did you use the Midpoint Formula?
Did you use the Midpoint Formula correctly?
Correct!
Do you subtract and then divide by two?
PTS: 1
DIF: Average
REF: Lesson 1-3
OBJ: 1-3.3 Find the midpoint of a segment.
NAT: NCTM ME.1
TOP: Find the midpoint of a segment.
KEY: Midpoint | Line Segment
15. ANS: B
Use a protractor. Align the 0 on either side of the scale with one side of the angle. Place the center point of the
protractor on the vertex. Find where the other side of the angle intersects the scale. Read the scale on which
you placed your 0.
Feedback
A
B
C
D
Read the other scale.
Correct!
Be more precise and read the other scale.
Be more precise.
PTS: 1
DIF: Basic
NAT: NCTM GM.1 | NCTM GM.1a
REF: Lesson 1-4
OBJ: 1-4.1 Measure angles.
TOP: Measure angles.
KEY: Angles | Measure Angles
16. ANS: A
Since
bisects
,
and
. Solve for w.
Feedback
A
B
C
D
Correct!
You are given the measure of FGH, not KGH.
You are not finding the measure of FGK. You are finding w.
Why did you divide by 2?
PTS:
OBJ:
TOP:
17. ANS:
1
DIF: Average
REF: Lesson 1-4
1-4.3 Identify and use congruent angles.
NAT: NCTM GM.1 | NCTM GM.1a
Identify and use congruent angles. KEY: Angles | Congruent Angles | Congruency
B
A ray bisects an angle. K is the endpoint of that ray, not P. The answer is
.
Feedback
A
B
C
D
Is a bisector a point?
Correct!
Is P the endpoint of that ray?
Is a bisector an angle?
PTS:
OBJ:
TOP:
18. ANS:
1
DIF: Basic
REF: Lesson 1-4
1-4.4 Identify and use the bisector of an angle.
Identify and use the bisector of an angle.
A
, so it is obtuse.
NAT: NCTM GM.1 | NCTM GM.1a
KEY: Angle Bisectors
Feedback
A
B
C
D
Correct!
If answer d is true, then this must be true.
Being in the interior means being between the two end rays of an angle.
If answer b is true, then this must be true.
PTS:
OBJ:
TOP:
19. ANS:
1
DIF: Basic
REF: Lesson 1-4
1-4.4 Identify and use the bisector of an angle.
Identify and use the bisector of an angle.
C
Feedback
A
B
That is the measure of JKM.
You forgot to add in x.
NAT: NCTM GM.1 | NCTM GM.1a
KEY: Angle Bisectors
C
D
Correct!
Opposite rays add up to 180.
PTS:
OBJ:
TOP:
20. ANS:
1
DIF: Average
REF: Lesson 1-4
1-4.4 Identify and use the bisector of an angle.
Identify and use the bisector of an angle.
D
NAT: NCTM GM.1 | NCTM GM.1a
KEY: Angle Bisectors
Feedback
A
B
C
D
Which angle are you looking for?
Did you solve for q instead of the angle measure?
Which two angles added together equal LKN?
Correct!
PTS:
OBJ:
TOP:
21. ANS:
1
DIF: Average
REF: Lesson 1-4
1-4.4 Identify and use the bisector of an angle.
Identify and use the bisector of an angle.
B
NAT: NCTM GM.1 | NCTM GM.1a
KEY: Angle Bisectors
Feedback
A
B
C
D
How many degrees in a straight angle?
Correct!
Did you solve for x instead of the angle measure?
How many degrees in a straight angle?
PTS: 1
DIF: Average
REF: Lesson 1-4
OBJ: 1-4.4 Identify and use the bisector of an angle.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify and use the bisector of an angle.
KEY: Angle Bisectors
22. ANS: B
Vertical angles are two nonadjacent angles formed by two intersecting lines. Acute angles measure less than
90 degrees.
Feedback
A
B
C
D
You are looking for vertical angles, not adjacent angles.
Correct!
You are looking for vertical angles, not a linear pair.
What is the definition of acute?
PTS:
OBJ:
TOP:
KEY:
1
DIF: Basic
REF: Lesson 1-5
1-5.1 Identify and use special pairs of angles.
NAT: NCTM GM.1 | NCTM GM.1a
Identify and use special pairs of angles.
Adjacent Angles | Vertical Angles | Linear Pair | Complementary Angles | Supplementary Angles
23. ANS: C
A linear pair is a pair of adjacent angles whose noncommon sides are opposite rays.
Feedback
A
B
C
D
You are looking for a linear pair, not vertical angles.
You are looking for a linear pair, not just adjacent angles.
Correct!
You are looking for a linear pair which, by definition, must be adjacent.
PTS: 1
DIF: Average
REF: Lesson 1-5
OBJ: 1-5.1 Identify and use special pairs of angles.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify and use special pairs of angles.
KEY: Adjacent Angles | Vertical Angles | Linear Pair | Complementary Angles | Supplementary Angles
24. ANS: A
Supplementary angles are two angles whose measures have a sum of 180.
Feedback
A
B
C
D
Correct!
What is the definition of supplementary?
Do the measures have a sum of 180 degrees?
What is the definition of supplementary?
PTS: 1
DIF: Basic
REF: Lesson 1-5
OBJ: 1-5.1 Identify and use special pairs of angles.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify and use special pairs of angles.
KEY: Adjacent Angles | Vertical Angles | Linear Pair | Complementary Angles | Supplementary Angles
25. ANS: D
Vertical angles are two nonadjacent angles formed by two intersecting lines. Obtuse angles measure greater
than 90 degrees.
Feedback
A
B
C
D
You are looking for vertical angles, not adjacent angles.
What is the definition of obtuse?
You are looking for vertical angles, not a linear pair.
Correct!
PTS: 1
DIF: Basic
REF: Lesson 1-5
OBJ: 1-5.1 Identify and use special pairs of angles.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify and use special pairs of angles.
KEY: Adjacent Angles | Vertical Angles | Linear Pair | Complementary Angles | Supplementary Angles
26. ANS: A
Complementary angles are two angles whose measures have a sum of 90.
Feedback
A
B
C
D
Correct!
Is that the value of q, or the measure of the angles?
What is the definition of complementary?
Is the sum of those angles 90?
PTS: 1
DIF: Average
REF: Lesson 1-5
OBJ: 1-5.1 Identify and use special pairs of angles.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify and use special pairs of angles.
KEY: Adjacent Angles | Vertical Angles | Linear Pair | Complementary Angles | Supplementary Angles
27. ANS: C
Complementary angles are two angles whose measures have a sum of 90.
Feedback
A
B
C
D
What is the definition of complementary?
Did you find the measures of two angles?
Correct!
Did you find y or the measures of the angles?
PTS: 1
DIF: Average
REF: Lesson 1-5
OBJ: 1-5.1 Identify and use special pairs of angles.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify and use special pairs of angles.
KEY: Adjacent Angles | Vertical Angles | Linear Pair | Complementary Angles | Supplementary Angles
28. ANS: B
Count the number of sides to determine the name of the polygon.
Feedback
A
B
C
D
How many sides does a hexagon have?
Correct!
How many sides does a pentagon have?
Count the number of sides.
PTS: 1
DIF: Average
REF: Lesson 1-6
OBJ: 1-6.1 Identify polygons.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify polygons.
KEY: Polygons | Identify Polygons
29. ANS: A
Count the number of sides to determine the name of the polygon.
Feedback
A
B
C
D
Correct!
Count the number of sides.
How many sides does a hexagon have?
How many sides does a decagon have?
PTS: 1
DIF: Average
REF: Lesson 1-6
OBJ: 1-6.1 Identify polygons.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify polygons.
KEY: Polygons | Identify Polygons
30. ANS: C
Suppose the line containing each side is drawn. If any of the lines contain any point in the interior of the
polygon, then it is concave. Otherwise it is convex.
A convex polygon in which all the sides are congruent and all the angles are congruent is called a regular
polygon.
Feedback
A
Count the number of sides.
B
C
D
If it is concave, lines drawn from the segments would pass through the polygon.
Correct!
If it is irregular, the angles and sides would not all be congruent.
PTS: 1
DIF: Basic
REF: Lesson 1-6
OBJ: 1-6.2 Name polygons.
NAT: NCTM ME.2
TOP: Name polygons.
KEY: Polygons | Name Polygons
31. ANS: D
Suppose the line containing each side is drawn. If any of the lines contain any point in the interior of the
polygon, then it is concave. Otherwise it is convex.
A convex polygon in which all the sides are congruent and all the angles are congruent is called a regular
polygon.
Feedback
A
B
C
D
If it is regular, the angles and sides would all be congruent.
If it is concave, lines drawn from the segments would pass through the polygon.
Count the number of sides.
Correct!
PTS: 1
DIF: Average
REF: Lesson 1-6
OBJ: 1-6.2 Name polygons.
NAT: NCTM ME.2
TOP: Name polygons.
KEY: Polygons | Name Polygons
32. ANS: D
Suppose the line containing each side is drawn. If any of the lines contain any point in the interior of the
polygon, then it is concave. Otherwise it is convex.
A convex polygon in which all the sides are congruent and all the angles are congruent is called a regular
polygon.
Feedback
A
B
C
D
If it is regular the angles and sides would all be congruent.
If it is concave, lines drawn from the segments would pass through the polygon.
Count the number of sides.
Correct!
PTS: 1
DIF: Average
NAT: NCTM ME.2
KEY: Polygons | Name Polygons
33. ANS: A
Perimeter is the sum of the sides.
REF: Lesson 1-6
OBJ: 1-6.2 Name polygons.
TOP: Name polygons.
Feedback
A
B
C
D
Correct!
Check your math.
Did you add all three sides?
What is the sum of the three sides?
PTS: 1
DIF: Average
REF: Lesson 1-6
OBJ: 1-6.3 Find perimeter of two-dimensional figures.
NAT: NCTM PS.1 | NCTM PS.2 | NCTM PS.3
KEY: Perimeter | Polygons
34. ANS: D
Perimeter is the sum of the sides.
TOP: Find the perimeters of polygons.
Feedback
A
B
C
D
What is the value of t?
Did you write and solve an equation?
Check your math.
Correct!
PTS:
OBJ:
NAT:
KEY:
35. ANS:
1
DIF: Average
REF: Lesson 1-6
1-6.3 Find perimeter of two-dimensional figures.
NCTM PS.1 | NCTM PS.2 | NCTM PS.3
Perimeter | Polygons
A
TOP: Find the perimeters of polygons.
The circumference C of a circle is the distance around the circle.
Feedback
A
B
C
D
Correct!
You need to multiply by 2.
Use the correct formula.
You have calculated the area.
PTS:
NAT:
KEY:
36. ANS:
1
DIF: Average
NCTM GM.1
Circumference | Circles
A
REF: Lesson 1-6
OBJ: 1-6.4 Find circumference of circles.
TOP: Find circumference of circles.
The area of a rectangle is the product of its length and width.
Feedback
A
B
C
D
Correct!
You need to multiply not add.
Place the decimal in the correct position.
You have calculated the perimeter.
PTS:
OBJ:
TOP:
37. ANS:
1
DIF: Basic
REF: Lesson 1-6
1-6.5 Find area of two-dimensional figures.
Find area of two-dimensional figures.
A
NAT: NCTM GM.1
KEY: Area | Two-Dimensional Figures
The area of a circle is the number of square units needed to cover a surface.
Feedback
A
B
C
D
Correct!
Use the correct formula.
You have calculated the circumference.
You have multiplied by an additional term.
PTS: 1
DIF: Basic
REF: Lesson 1-6
OBJ: 1-6.5 Find area of two-dimensional figures.
NAT: NCTM GM.1
TOP: Find area of two-dimensional figures.
KEY: Area | Two-Dimensional Figures
38. ANS: B
The set of all points in space that are a given distance from a given point is called a sphere. It is the threedimensional equivalent of a circle.
Feedback
A
B
C
D
A cone has a circular base and a vertex.
Correct!
A cylinder has two circular bases.
A prism has two congruent polygons for bases.
PTS: 1
DIF: Basic
REF: Lesson 1-7
OBJ: 1-7.1 Identify three-dimensional figures.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify three-dimensional figures. KEY: Three-Dimensional Figures
39. ANS: C
This solid is a cylinder. The bases of a cylinder are circles. In this cylinder, the circular bases are centered at
points A and B, thus the correct answer is
and
.
Feedback
A
B
C
D
The bases of a cylinder are circles, not segments.
The bases of a cylinder are circles, not segments.
Correct!
Those are the wrong centers for the circular bases.
PTS: 1
DIF: Basic
REF: Lesson 1-7
OBJ: 1-7.1 Identify three-dimensional figures.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify three-dimensional figures. KEY: Three-Dimensional Figures
40. ANS: A
The edges of a solid are the line segments formed by the intersection of the faces. This figure has 12 such
segments: four on top, four in the middle and four on the bottom.
Feedback
A
B
C
D
Correct!
Those are only the edges on the top of the solid. There are eight more.
Those are only the edges on the bottom of the solid. There are eight more.
Those are only the edges in the middle of the solid. There are eight more.
PTS:
OBJ:
TOP:
41. ANS:
1
DIF: Basic
REF: Lesson 1-7
1-7.1 Identify three-dimensional figures.
Identify and use three-dimensional figures.
A
NAT: NCTM GM.1 | NCTM GM.1a
KEY: Three-Dimensional Figures
A face of a solid is a flat surface. This solid contains five such surfaces: three lateral rectangles and two
triangular bases.
Feedback
A
B
C
D
Correct!
There is one more face on the bottom of the solid.
The bases of the solid are also faces.
There are three more faces besides the two base triangles.
PTS: 1
DIF: Basic
REF: Lesson 1-7
OBJ: 1-7.1 Identify three-dimensional figures.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify three-dimensional figures. KEY: Three-Dimensional Figures
42. ANS: A
Start with 1. Add, subtract, or multiply the same number to each number to get the next one.
Feedback
A
B
C
D
Correct!
What operations are involved?
Didn’t you carry the conjecture too far?
Check your math.
PTS: 1
DIF: Basic
REF: Lesson 2-1
OBJ: 2-1.1 Make conjectures based on inductive reasoning.
NAT: NCTM RP.2
TOP: Make conjectures based on inductive reasoning.
KEY: Inductive Reasoning | Conjectures
43. ANS: A
Concave polygons must be irregular. This means all sides and angles are not congruent.
Feedback
A
B
C
D
Correct!
What is the definition of concave?
Is that counterexample correct?
What is the definition of concave?
PTS: 1
DIF: Basic
REF: Lesson 2-1
OBJ: 2-1.2 Find counterexamples.
NAT: NCTM RP.3 TOP: Find counterexamples.
KEY: Counterexamples
44. ANS: C
Angles are congruent only if their measures are equal. Point B may be closer to line AD or line DC so the
measures would not be equal.
Feedback
A
B
C
D
What is the definition of congruent?
What is the definition of congruent?
Correct!
Would that be a counterexample?
PTS: 1
DIF: Basic
REF: Lesson 2-1
OBJ: 2-1.2 Find counterexamples.
NAT: NCTM RP.3 TOP: Find counterexamples.
KEY: Counterexamples
45. ANS: D
Because m is squared in the example, m could be positive or negative.
Feedback
A
B
C
D
Subtract 6 from both sides.
What about negative numbers?
Subtract 6 from both sides.
Correct!
PTS: 1
DIF: Basic
REF: Lesson 2-1
OBJ: 2-1.2 Find counterexamples.
NAT: NCTM RP.3 TOP: Find counterexamples.
KEY: Counterexamples
46. ANS: B
Coplanar points always lie in the same plane. Three points are always coplanar but four are not.
Feedback
A
B
C
D
What does coplanar mean?
Correct!
What does coplanar mean?
Would the points have to be in the same plane?
PTS: 1
DIF: Basic
REF: Lesson 2-1
OBJ: 2-1.2 Find counterexamples.
NAT: NCTM RP.3 TOP: Find counterexamples.
KEY: Counterexamples
47. ANS: C
If two angles are supplementary their measures total 180. Either both are right or one is obtuse and the other
acute.
Feedback
A
B
C
D
What is the definition of supplementary?
What is the definition of supplementary?
Correct!
What is the definition of supplementary?
PTS: 1
DIF: Basic
REF: Lesson 2-1
OBJ: 2-1.2 Find counterexamples.
NAT: NCTM RP.3 TOP: Find counterexamples.
KEY: Counterexamples
48. ANS: D
Unless there are specific angle measures mentioned, even though the angles in the picture may look congruent
you cannot assume that they are congruent.
Feedback
A
B
C
D
What is the definition of congruent?
What is the definition of congruent?
What is the definition of congruent?
Correct!
PTS: 1
DIF: Basic
REF: Lesson 2-1
OBJ: 2-1.2 Find counterexamples.
NAT: NCTM RP.3 TOP: Find counterexamples.
KEY: Counterexamples
49. ANS: B
Even though they have a common point, the two segments do not have to be on the same line.
Feedback
A
What is the definition of midpoint?
B
C
D
Correct!
What is the definition of midpoint?
What is the definition of midpoint?
PTS: 1
DIF: Basic
REF: Lesson 2-1
OBJ: 2-1.2 Find counterexamples.
NAT: NCTM RP.3 TOP: Find counterexamples.
KEY: Counterexamples
50. ANS: A
Two or more statements can be joined to form a compound statement. A conjunction is a compound statement
formed by joining two or more statements with the word and.
Feedback
A
B
C
D
Correct!
What does statement r say?
Are four points always coplanar?
Does a decagon have 12 sides?
PTS: 1
DIF: Basic
REF: Lesson 2-2
OBJ: 2-2.1 Determine truth values of conjunctions and disjunctions.
NAT: NCTM RP.3 TOP: Determine truth values of conjunctions and disjunctions.
KEY: Truth Values | Conjunctions | Disjunctions
51. ANS: D
Two or more statements can be joined to form a compound statement. A conjunction is a compound statement
formed by joining two or more statements with the word and. A disjunction is a compound statement formed
by joining two or more statements with the word or. The symbol for logical and
is . The symbol for logical or is .
Feedback
A
B
C
D
What is the symbol for logical and?
Are four points always coplanar?
What is the symbol for logical and?
Correct!
PTS: 1
DIF: Average
REF: Lesson 2-2
OBJ: 2-2.1 Determine truth values of conjunctions and disjunctions.
NAT: NCTM RP.3 TOP: Determine truth values of conjunctions and disjunctions.
KEY: Truth Values | Conjunctions | Disjunctions
52. ANS: B
Two or more statements can be joined to form a compound statement. A conjunction is a compound statement
formed by joining two or more statements with the word and. A disjunction is a compound statement formed
by joining two or more statements with the word or. The symbol for logical and is . The symbol for logical
or is .
Feedback
A
B
C
D
What is the symbol for logical or?
Correct!
How many sides does a decagon have?
What is the symbol for logical or?
PTS: 1
DIF: Average
REF: Lesson 2-2
OBJ: 2-2.1 Determine truth values of conjunctions and disjunctions.
NAT: NCTM RP.3 TOP: Determine truth values of conjunctions and disjunctions.
KEY: Truth Values | Conjunctions | Disjunctions
53. ANS: A
Two or more statements can be joined to form a compound statement. A conjunction is a compound statement
formed by joining two or more statements with the word and. A disjunction is a compound statement formed
by joining two or more statements with the word or. The symbol for logical and is . The symbol for logical
or is .
Feedback
A
B
C
D
Correct!
What is the measure of a right angle?
How many sides does a decagon have?
What is the measure of a right angle?
PTS: 1
DIF: Average
REF: Lesson 2-2
OBJ: 2-2.1 Determine truth values of conjunctions and disjunctions.
NAT: NCTM RP.3 TOP: Determine truth values of conjunctions and disjunctions.
KEY: Truth Values | Conjunctions | Disjunctions
54. ANS: D
Two or more statements can be joined to form a compound statement. A conjunction is a compound statement
formed by joining two or more statements with the word and. A disjunction is a compound statement formed
by joining two or more statements with the word or. The symbol for logical and is . The symbol for logical
or is .
Feedback
A
B
C
D
How many sides does a decagon have?
What is the symbol for logical and?
What is the symbol for logical and?
Correct!
PTS: 1
DIF: Average
REF: Lesson 2-2
OBJ: 2-2.1 Determine truth values of conjunctions and disjunctions.
NAT: NCTM RP.3 TOP: Determine truth values of conjunctions and disjunctions.
KEY: Truth Values | Conjunctions | Disjunctions
55. ANS: A
Two or more statements can be joined to form a compound statement. A conjunction is a compound statement
formed by joining two or more statements with the word and. A disjunction is a compound statement formed
by joining two or more statements with the word or. The symbol for logical and is . The symbol for logical
or is .
Feedback
A
B
C
D
Correct!
Are four points always coplanar?
What is the symbol for logical and?
What is the symbol for logical and?
PTS: 1
DIF: Average
REF: Lesson 2-2
OBJ: 2-2.1 Determine truth values of conjunctions and disjunctions.
NAT: NCTM RP.3 TOP: Determine truth values of conjunctions and disjunctions.
KEY: Truth Values | Conjunctions | Disjunctions
56. ANS: C
The first statement column in a truth table contains half Ts, half Fs, grouped together. The second statement
column in a truth table contains the same, but they are grouped by half the number that the first column was.
The third statement column contains the same but they are grouped by half the number that the second
column was. Use the truth values of the first three columns to determine the truth values for the last three
columns. The symbol for not is . The symbol for logical and is .
Feedback
A
B
C
D
Check the values for the last column carefully.
Check the values for the last column carefully.
Correct!
Do your statement columns show every possible T and F combination?
PTS: 1
DIF: Average
REF: Lesson 2-2
OBJ: 2-2.2 Construct truth tables.
NAT: NCTM RP.3 TOP: Construct truth tables. KEY:
Truth Tables
57. ANS: C
The inverse is negating both the hypothesis and conclusion of the conditional.
Feedback
A
B
C
D
Remember
.
Remember
.
Correct!
Is that the converse?
PTS: 1
DIF: Average
REF: Lesson 2-3
OBJ: 2-3.3 Write the inverse of if-then statements.
NAT: NCTM RP.3
TOP: Write the inverse of if-then statements.
KEY: Inverse | If-Then Statements
58. ANS: D
The inverse is negating both the hypothesis and conclusion of the conditional.
Feedback
A
B
C
D
Is that the converse?
Remember
.
Remember
.
Correct!
PTS: 1
DIF: Average
REF: Lesson 2-3
OBJ: 2-3.3 Write the inverse of if-then statements.
NAT: NCTM RP.3
TOP: Write the inverse of if-then statements.
KEY: Inverse | If-Then Statements
59. ANS: B
The inverse is negating both the hypothesis and conclusion of the conditional.
Feedback
A
B
Remember
Correct!
.
C
D
Remember
Remember
.
.
PTS: 1
DIF: Average
REF: Lesson 2-3
OBJ: 2-3.3 Write the inverse of if-then statements.
NAT: NCTM RP.3
TOP: Write the inverse of if-then statements.
KEY: Inverse | If-Then Statements
60. ANS: A
In the contrapositive you negate both the hypothesis and conclusion of the converse statement.
Feedback
A
B
C
D
Correct!
Is that the inverse?
Remember
Remember
.
.
PTS: 1
DIF: Basic
REF: Lesson 2-3
OBJ: 2-3.4 Write the contrapositive of if-then statements.
NAT: NCTM RP.1
TOP: Write the contrapositive of if-then statements.
KEY: Contrapositive | If-Then Statements
61. ANS: D
In the contrapositive you negate both the hypothesis and conclusion of the converse statement.
Feedback
A
B
C
D
Remember
Remember
Remember
Correct!
.
.
.
PTS: 1
DIF: Average
REF: Lesson 2-3
OBJ: 2-3.4 Write the contrapositive of if-then statements.
NAT: NCTM RP.1
TOP: Write the contrapositive of if-then statements.
KEY: Contrapositive | If-Then Statements
62. ANS: C
In the contrapositive you negate both the hypothesis and conclusion of the converse statement.
Feedback
A
B
C
D
Is this true?
Remember
Correct!
Remember
.
.
PTS: 1
DIF: Average
REF: Lesson 2-3
OBJ: 2-3.4 Write the contrapositive of if-then statements.
NAT: NCTM RP.1
TOP: Write the contrapositive of if-then statements.
KEY: Contrapositive | If-Then Statements
63. ANS: B
Postulates:
1. Through any two points, there is exactly one line.
2. Through any three points not on the same line, there is exactly one plane.
3. A line contains at least two points.
4. A plane contains at least three points not on the same line.
5. If two points lie in a plane, then the entire line containing those points lies in that plane.
6. If two lines intersect, then their intersection is exactly one point.
7. If two planes intersect, then their intersection is a line.
Feedback
A
B
C
D
Does that apply?
Correct!
Is that a postulate?
Does that fit the situation?
PTS: 1
DIF: Average
REF: Lesson 2-5
OBJ: 2-5.1 Identify and use basic postulates about points, lines, and planes.
NAT: NCTM RP.1 TOP: Identify and use basic postulates about points, lines, and planes.
KEY: Points | Lines | Planes
64. ANS: C
Postulates:
1. Through any two points, there is exactly one line.
2. Through any three points not on the same line, there is exactly one plane.
3. A line contains at least two points.
4. A plane contains at least three points not on the same line.
5. If two points lie in a plane, then the entire line containing those points lies in that plane.
6. If two lines intersect, then their intersection is exactly one point.
7. If two planes intersect, then their intersection is a line.
Feedback
A
B
C
D
Is that a postulate?
Does that fit the situation?
Correct!
Does that apply?
PTS: 1
DIF: Average
REF: Lesson 2-5
OBJ: 2-5.1 Identify and use basic postulates about points, lines, and planes.
NAT: NCTM RP.1 TOP: Identify and use basic postulates about points, lines, and planes.
KEY: Points | Lines | Planes
65. ANS: A
Planes that intersect have a common line.
Feedback
A
B
C
D
Correct!
This plane has four lines to intersect with other planes.
Do they all intersect plane CDI in a line?
This plane has four lines to intersect with other planes.
PTS: 1
DIF: Basic
REF: Lesson 3-1
OBJ: 3-1.1 Identify the relationships between two lines or two planes.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify the relationships between two lines or two planes.
KEY: Relationship Between Two Lines | Relationship Between Two Planes
66. ANS: B
Coplanar segments that do not intersect are parallel.
Feedback
A
B
C
D
Are those parallel to
?
Correct!
Are those all of the segments parallel to
Are those all of the segments parallel to
?
?
PTS: 1
DIF: Basic
REF: Lesson 3-1
OBJ: 3-1.1 Identify the relationships between two lines or two planes.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify the relationships between two lines or two planes.
KEY: Relationship Between Two Lines | Relationship Between Two Planes
67. ANS: B
Planes intersect in a line.
Feedback
A
B
C
D
Do they all intersect plane CHG in a line?
Correct!
Is that all?
Do they all intersect plane CHG in a line?
PTS: 1
DIF: Average
REF: Lesson 3-1
OBJ: 3-1.1 Identify the relationships between two lines or two planes.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify the relationships between two lines or two planes.
KEY: Relationship Between Two Lines | Relationship Between Two Planes
68. ANS: A
Skew lines do not intersect and are not coplanar.
Feedback
A
B
C
D
Correct!
Skew lines do not intersect.
Skew lines are not coplanar.
Skew lines are not coplanar.
PTS: 1
DIF: Average
REF: Lesson 3-1
OBJ: 3-1.1 Identify the relationships between two lines or two planes.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify the relationships between two lines or two planes.
KEY: Relationship Between Two Lines | Relationship Between Two Planes
69. ANS: A
Skew lines do not intersect and are not coplanar.
Feedback
A
B
C
D
Correct!
Skew lines do not intersect.
Skew lines are not coplanar.
Skew lines are not coplanar.
PTS:
OBJ:
NAT:
KEY:
1
DIF: Average
REF: Lesson 3-1
3-1.1 Identify the relationships between two lines or two planes.
NCTM GM.1 | NCTM GM.1a
TOP: Identify the relationships between two lines or two planes.
Relationship Between Two Lines | Relationship Between Two Planes
70. ANS: B
Planes intersect in a line.
Feedback
A
B
C
D
Do they all intersect plane CHG in a line?
Correct!
Is that all?
Do they all intersect plane CHG in a line?
PTS: 1
DIF: Average
REF: Lesson 3-1
OBJ: 3-1.1 Identify the relationships between two lines or two planes.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify the relationships between two lines or two planes.
KEY: Relationship Between Two Lines | Relationship Between Two Planes
71. ANS: C
Coplanar segments that do not intersect are parallel.
Feedback
A
B
C
D
Parallel lines do not intersect.
Parallel lines are coplanar.
Correct!
Those segments are parallel to which line?
PTS: 1
DIF: Basic
REF: Lesson 3-1
OBJ: 3-1.1 Identify the relationships between two lines or two planes.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Identify the relationships between two lines or two planes.
KEY: Relationship Between Two Lines | Relationship Between Two Planes
72. ANS: D
A line that intersects two or more lines in a plane at different points is called a transversal.
Feedback
A
B
C
D
What about line p?
You need every combination of the lines.
What is the definition of transversal?
Correct!
PTS: 1
DIF: Basic
REF: Lesson 3-1
OBJ: 3-1.2 Name angles formed by a pair of lines and a transversal.
NAT: NCTM GM.1 | NCTM GM.1b
TOP: Name angles formed by a pair of lines and a transversal.
KEY: Transversals | Two Lines and a Transversal | Angles
73. ANS: A
A line that intersects two or more lines in a plane at different points is called a transversal.
Feedback
A
B
C
D
Correct!
What is the definition of transversal?
You need every combination of lines.
You need every combination of the lines.
PTS: 1
DIF: Average
REF: Lesson 3-1
OBJ: 3-1.2 Name angles formed by a pair of lines and a transversal.
NAT: NCTM GM.1 | NCTM GM.1b
TOP: Name angles formed by a pair of lines and a transversal.
KEY: Transversals | Two Lines and a Transversal | Angles
74. ANS: D
Extend v to intersect with p. This creates a linear pair at point S with angles measuring
 (given) and 61.
The angles formed by the intersection of v and p (also linear pairs) measure
 (corresponding angles) and
55 with the latter being one of the interior angles of the triangle formed by t, p, and v. Since the sum of the
angles of a triangle is 180, the angle that is vertical to 1 is 64, thus making 1 64 as well.
Feedback
A
B
C
D
Extend v as a transversal of q and p.
Extend v as a transversal of q and p.
Extend v as a transversal of q and p.
Correct!
PTS:
OBJ:
NAT:
KEY:
75. ANS:
1
DIF: Average
REF: Lesson 3-2
3-2.2 Use algebra to find angle measures.
NCTM AL.4a | NCTM GM.2 | NCTM GM.2a
Angles | Angle Measures
C
The formula for slope is
TOP: Use algebra to find angle measures.
. If the slopes are the same, the lines are parallel. If the product of the two
slopes is –1, the lines are perpendicular.
Feedback
A
B
C
Parallel slopes are the same and perpendicular ones are opposite reciprocals.
Parallel slopes are the same and perpendicular ones are opposite reciprocals.
Correct!
PTS: 1
DIF: Average
REF: Lesson 3-3
OBJ: 3-3.2 Use slope to identify parallel and perpendicular lines.
NAT: NCTM AL.2 | NCTM AL.2c | NCTM RE.2
TOP: Use slope to identify parallel lines and perpendicular lines.
KEY: Parallel Lines | Perpendicular Lines | Slope
76. ANS: B
The slope-intercept form is
. Point b is the point at which
.
Feedback
A
B
C
D
What is the y-intercept?
Correct!
What is the slope of the line?
Remember y = mx + b.
PTS:
OBJ:
NAT:
TOP:
KEY:
1
DIF: Average
REF: Lesson 3-4
3-4.1 Write an equation of a line given information about its graph.
NCTM AL.2
Write an equation of a line given information about its graph.
Equation of Lines | Graphs
77. ANS: D
The point-slope form is
. Point
is a point through which the line passes.
Feedback
A
B
C
D
Is that point-slope form?
What is the slope?
Remember the point is (x1, y1).
Correct!
PTS: 1
DIF: Average
REF: Lesson 3-4
OBJ: 3-4.2 Solve problems by writing equations.
NAT: NCTM GM.1 | NCTM GM.1b
TOP: Solve problems by writing equations.
KEY: Solve Problems | Write Equations
78. ANS: A
The point-slope form is
. Point
is a point through which the line passes.
Feedback
A
B
C
D
Correct!
What are x1 and y1?
Be careful with sign rules?
Is that point-slope form?
PTS: 1
DIF: Average
REF: Lesson 3-4
OBJ: 3-4.2 Solve problems by writing equations.
NAT: NCTM GM.1 | NCTM GM.1b
TOP: Solve problems by writing equations.
KEY: Solve Problems | Write Equations
79. ANS: C
The point-slope form is
. Point
is a point through which the line passes.
Feedback
A
B
C
D
Is that point-slope form?
Remember the point is (x1, y1).
Correct!
Be careful with signs.
PTS: 1
DIF: Average
REF: Lesson 3-4
OBJ: 3-4.2 Solve problems by writing equations.
NAT: NCTM GM.1 | NCTM GM.1b
TOP: Solve problems by writing equations.
KEY: Solve Problems | Write Equations
80. ANS: C
Postulates and theorems:
If corresponding angles are congruent, then lines are parallel.
If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to
the given line.
If alternate exterior angles are congruent, then lines are parallel.
If consecutive interior angles are supplementary, then lines are parallel.
If alternate interior angles are congruent, then lines are parallel.
If two lines are perpendicular to the same line, then lines are parallel.
Feedback
A
What kind of angles are those?
B
C
D
What kind of angles are those?
Correct!
Which lines are parallel?
PTS: 1
DIF: Average
REF: Lesson 3-5
OBJ: 3-5.1 Recognize angle conditions that occur with parallel lines.
NAT: NCTM GM.1b | NCTM GM.1c | NCTM RP.3
TOP: Recognize angle conditions that occur with parallel lines.
KEY: Angles | Parallel Lines
81. ANS: B
The slope of a line p perpendicular to m has the negative reciprocal to the equation of line m.
The slope of line m is
vertical line,
, which is 0. This is a horizontal line. The perpendicular line, then would be a
, going through point P. The point on m where line p intersects it would be
. Use the
Distance Formula to find the distance from point P to the point on m that intersects line p.
Feedback
A
B
C
D
Remember it is (x, y).
Correct!
You want the distance to line m.
You want the distance to line m.
PTS: 1
DIF: Basic
REF: Lesson 3-6
OBJ: 3-6.1 Find the distance between a point and a line.
NAT: NCTM GM.2 | NCTM GM.2a
TOP: Find the distance between a point and a line.
KEY: Distance | Distance Between a Point and a Line
82. ANS: C
The slope of a line perpendicular to each has the negative reciprocal to the equation of one of the lines. The
perpendicular line containing the y-intercept would be
. You need to find a common point for the
two lines, so set the equations equal to each other.
Use the Distance Formula from that common point to the y-intercept of the first line.
Feedback
A
B
C
D
Construct a perpendicular line and find the intersection point on each parallel line.
Construct a perpendicular line and find the intersection point on each parallel line.
Correct!
Construct a perpendicular line and find the intersection point on each parallel line.
PTS: 1
DIF: Average
REF: Lesson 3-6
OBJ: 3-6.2 Find the distance between parallel lines.
NAT: NCTM PS.1 | NCTM PS.2 | NCTM PS.3
TOP: Find the distance between parallel lines.
KEY: Distance | Parallel Lines | Distance Between Parallel Lines
83. ANS: B
An acute triangle has 3 acute angles.
An obtuse triangle has one obtuse angle.
A right triangle has one right angle.
An equiangular triangle has at least 2 congruent sides.
Feedback
A
B
C
D
Check for congruent sides and measure angles.
Correct!
Check for congruent sides and measure angles.
Check for congruent sides and measure angles.
PTS: 1
DIF: Basic
REF: Lesson 4-1
OBJ: 4-1.1 Identify and classify triangles by angles.
TOP: Identify and classify triangles by angles.
84. ANS: D
Use the Distance Formula to find the lengths of the sides.
NAT: NCTM GM.1 | NCTM GM.1a
KEY: Triangles | Classify Triangles
If
or
or
, then the triangle is isosceles.
If
, then the triangle is equilateral.
If neither of the above, the triangle is scalene.
Feedback
A
B
C
D
Use the distance formula to find the lengths of the sides.
What are the lengths of the sides?
Did you use the distance formula?
Correct!
PTS: 1
DIF: Average
REF: Lesson 4-1
OBJ: 4-1.2 Identify and classify triangles by sides.
TOP: Identify and classify triangles by sides.
NAT: NCTM GM.1 | NCTM GM.1b
KEY: Triangles | Classify Triangles
85. ANS: B
The Angle Sum Theorem states that the sum of the measures of the angles of a triangle is 180.
Feedback
A
B
C
D
Did you use the Angle Sum Theorem.
Correct!
Use the Angle Sum Theorem.
Use the Angle Sum Theorem.
PTS: 1
DIF: Average
REF: Lesson 4-2
OBJ: 4-2.1 Apply the Angle Sum Theorem.
NAT: NCTM GM.1 | NCTM GM.1b
TOP: Apply the Angle Sum Theorem.
KEY: Angle Sum Theorem
86. ANS: A
The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of
the measures of the two remote interior angles.
Feedback
A
B
C
D
Correct!
What is the sum of the measures of the angles in a triangle?
Did you use the Exterior Angle Theorem?
Use the Exterior Angle Theorem.
PTS: 1
DIF: Average
REF: Lesson 4-2
OBJ: 4-2.2 Apply the Exterior Angle Theorem.
NAT: NCTM GM.1 | NCTM GM.1b
TOP: Apply the Exterior Angle Theorem.
KEY: Exterior Angle Theorem
87. ANS: D
The letters naming the triangles correspond to the congruent vertices of the two triangles in the same order.
Feedback
A
B
C
D
Be careful with the order of the vertices.
Are the vertices in the correct order?
The letters naming the triangles correspond to the congruent vertices of the two
triangles.
Correct!
PTS: 1
DIF: Average
REF: Lesson 4-3
OBJ: 4-3.2 Identify congruent transformations.
NAT: NCTM GM.1 | NCTM GM.1b
TOP: Identify congruent transformations.
KEY: Transformations | Congruence Transformations
88. ANS: D
If each side of triangle PQR is the same length as the corresponding side of triangle STU, then the triangles
are congruent.
Feedback
A
B
C
D
Use the SSS Postulate.
Check your math.
How do you determine if two triangles are congruent?
Correct!
PTS:
OBJ:
NAT:
KEY:
89. ANS:
1
DIF: Average
REF: Lesson 4-4
4-4.1 Use the SSS Postulate to test for triangle congruence.
NCTM GM.1 | NCTM GM.1b
TOP: Use the SSS Postulate to test for triangle congruence.
SSS Postulate | Congruent Triangles
A
Feedback
A
B
C
D
Correct!
What do you know about the sides of an equilateral triangle?
How many degrees is each angle of an equilateral triangle?
Did you add or subtract when solving for y?
PTS:
OBJ:
TOP:
90. ANS:
1
DIF: Average
REF: Lesson 4-6
4-6.2 Use the properties of equilateral triangles.
Use the properties of equilateral triangles.
D
NAT: NCTM GM.2 | NCTM GM.2a
KEY: Equilateral Triangles
Feedback
A
B
C
D
Did you use the Pythagorean Theorem?
Did you add instead of subtract?
Check your math.
Correct!
PTS:
OBJ:
TOP:
91. ANS:
1
DIF: Average
REF: Lesson 4-6
4-6.2 Use the properties of equilateral triangles.
Use the properties of equilateral triangles.
D
NAT: NCTM GM.2 | NCTM GM.2a
KEY: Equilateral Triangles
Feedback
A
B
C
D
Did you use the Pythagorean Theorem correctly?
Check your math.
Should you have subtracted?
Correct!
PTS:
OBJ:
TOP:
92. ANS:
OBJ:
TOP:
93. ANS:
1
DIF: Average
REF: Lesson 4-6
4-6.2 Use the properties of equilateral triangles.
NAT: NCTM GM.2 | NCTM GM.2a
Use the properties of equilateral triangles.
KEY: Equilateral Triangles
B
PTS: 1
DIF: Basic
REF: Lesson 4-7
4-7.1 Identify reflections, translations, and rotations.
NAT: NCTM GM.2 | NCTM GM.3
Congruence transformations.
KEY: congruence transformation
B
1. Use the origin as a vertex or center of the figure.
2. Place at least one side of a polygon on an axis.
3. Keep the figure within the first quadrant if possible.
4. Use coordinates that make computations as simple as possible.
Feedback
A
B
C
D
Look at your labels.
Correct!
Which sides are congruent?
Which sides are congruent?
PTS: 1
DIF: Basic
REF: Lesson 4-8
OBJ: 4-8.1 Position and label triangles for use in coordinate proofs.
NAT: NCTM PS.1 | NCTM PS.2 | NCTM PS.3
TOP: Position and label triangles for use in coordinate proofs. KEY: Proofs | Coordinate Proofs
94. ANS: A
1. Use the origin as a vertex or center of the figure.
2. Place at least one side of a polygon on an axis.
3. Keep the figure within the first quadrant if possible.
4. Use coordinates that make computations as simple as possible.
Feedback
A
B
C
D
Correct!
What is the base?
What is the base of that triangle?
Look at your labels.
PTS: 1
DIF: Average
REF: Lesson 4-8
OBJ: 4-8.1 Position and label triangles for use in coordinate proofs.
NAT: NCTM PS.1 | NCTM PS.2 | NCTM PS.3
TOP: Position and label triangles for use in coordinate proofs. KEY: Proofs | Coordinate Proofs
95. ANS: D
1. Use the origin as a vertex or center of the figure.
2. Place at least one side of a polygon on an axis.
3. Keep the figure within the first quadrant if possible.
4. Use coordinates that make computations as simple as possible.
Feedback
A
B
C
D
What segment is the hypotenuse?
Look at your labels.
Which leg is 4a units long?
Correct!
PTS: 1
DIF: Average
REF: Lesson 4-8
OBJ: 4-8.1 Position and label triangles for use in coordinate proofs.
NAT: NCTM PS.1 | NCTM PS.2 | NCTM PS.3
TOP: Position and label triangles for use in coordinate proofs. KEY: Proofs | Coordinate Proofs
96. ANS: D
1. Use the origin as a vertex or center of the figure.
2. Place at least one side of a polygon on an axis.
3. Keep the figure within the first quadrant if possible.
4. Use coordinates that make computations as simple as possible.
Feedback
A
B
C
D
Which segment is the base?
Where does the triangle belong?
Look at your labels.
Correct!
PTS: 1
DIF: Basic
REF: Lesson 4-8
OBJ: 4-8.1 Position and label triangles for use in coordinate proofs.
NAT: NCTM PS.1 | NCTM PS.2 | NCTM PS.3
TOP: Position and label triangles for use in coordinate proofs. KEY: Proofs | Coordinate Proofs
97. ANS: A
1. Use the origin as a vertex or center of the figure.
2. Place at least one side of a polygon on an axis.
3. Keep the figure within the first quadrant if possible.
4. Use coordinates that make computations as simple as possible.
Feedback
A
B
C
D
Correct!
Look at your placement.
How tall is the triangle?
How tall is the triangle?
PTS: 1
DIF: Average
REF: Lesson 4-8
OBJ: 4-8.1 Position and label triangles for use in coordinate proofs.
NAT: NCTM PS.1 | NCTM PS.2 | NCTM PS.3
TOP: Position and label triangles for use in coordinate proofs. KEY: Proofs | Coordinate Proofs
98. ANS: A
If
is an altitude,
. The measures of the angles of every triangle add up to 180.
Feedback
A
B
C
D
Correct!
Which angle measures add up to 180?
Which angles must measure 90°?
Check your math.
PTS: 1
DIF: Average
REF: Lesson 5-2
OBJ: 5-2.1 Use altitudes in triangles.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Use altitudes in triangles.
KEY: Altitudes | Triangles
99. ANS: D
The sum of the lengths of any two sides must be greater than the third.
Feedback
A
B
C
D
Did you check all the sums?
Add two sides and compare to the third.
Add two sides and compare to the third.
Correct!
PTS: 1
DIF: Basic
REF: Lesson 5-5
OBJ: 5-5.1 Apply the Triangle Inequality Theorem.
NAT: NCTM GM.2 | NCTM GM.2a
TOP: Apply the Triangle Inequality Theorem.
KEY: Triangles Inequality Theorem
100. ANS: C
The sum of the lengths of any two sides must be greater than the third.
Feedback
A
B
C
D
Did you check all the sums?
Add two sides and compare to the third.
Correct!
Add two sides and compare to the third.
PTS: 1
DIF: Average
REF: Lesson 5-5
OBJ: 5-5.1 Apply the Triangle Inequality Theorem.
TOP: Apply the Triangle Inequality Theorem.
NAT: NCTM GM.2 | NCTM GM.2a
KEY: Triangles Inequality Theorem
SHORT ANSWER
101. ANS:
No; the average daily temperature for City G could be as low as
daily temperature for City H could be as low as
or as high as
could be as high as
or as high as
The average
The difference in temperature
An absolute error of 0.5 is acceptable. Add and subtract the absolute error from the given measurement to get
the high and low values respectively.
PTS: 1
DIF: Advanced
REF: Lesson 1-2
NAT: NCTM ME.2 | NCTM ME.2a | NCTM NO.1
KEY: Solve multi-step problems.
102. ANS:
OBJ: 1-2.4 Solve multi-step problems.
TOP: Solve multi-step problems.
The midpoint of a segment is the point halfway between the endpoints of the segment.
The coordinates of the midpoint of a segment with endpoints that have the coordinates
and
are
PTS: 1
DIF: Advanced
REF: Lesson 1-3
NAT: NCTM GM.2 | NCTM GM.2a | NCTM ME.1
KEY: Solve multi-step problems.
103. ANS:
62, 28
Calculate the angle of reflection by using the relation
.
Also, r and
are complementary i.e.
OBJ: 1-3.4 Solve multi-step problems.
TOP: Solve multi-step problems.
PTS: 1
DIF: Basic
NAT: NCTM GM.1 | NCTM GM.1a
KEY: Solve multi-step problems.
104. ANS:
REF: Lesson 1-5
OBJ: 1-5.2 Solve multi-step problems.
TOP: Solve multi-step problems.
Calculate the angle of reflection by using the relation
Also, i and
are complementary i.e.
PTS: 1
DIF: Advanced
NAT: NCTM GM.1 | NCTM GM.1a
KEY: Solve multi-step problems.
105. ANS:
3.8 in.
.
REF: Lesson 1-5
OBJ: 1-5.2 Solve multi-step problems.
TOP: Solve multi-step problems.
Here, the perimeter of the square is equal to the circumference of the circular loop. Use this relationship to
find the maximum radius of the loop.
PTS: 1
DIF: Advanced
REF: Lesson 1-6
OBJ: 1-6.6 Solve multi-step problems.
NAT: NCTM GM.1 | NCTM GM.1a | NCTM ME.2 | NCTM PS.1 | NCTM PS.2 | NCTM PS.3
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.
106. ANS:
19.1 cm
Here, the perimeter of the square painting is equal to the circumference of the circular picture frame. Use this
relationship to find the maximum radius of the frame.
PTS: 1
DIF: Average
REF: Lesson 1-6
OBJ: 1-6.6 Solve multi-step problems.
NAT: NCTM GM.1 | NCTM GM.1a | NCTM ME.2 | NCTM PS.1 | NCTM PS.2 | NCTM PS.3
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.
107. ANS:
52.7
The surface area T, of a triangular prism is the sum of the areas of each face of a solid.
; where P is the perimeter of the base, h is the height, and B is the area of the base.
PTS: 1
DIF: Advanced
REF: Lesson 1-7
OBJ: 1-7.4 Solve multi-step problems.
NAT: NCTM GM.1 | NCTM GM.1a | NCTM GM.4 | NCTM GM.4a | NCTM GM.4b | NCTM GM.1
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.
108. ANS:
Sample: Two adjacent angles formed by two intersecting lines are not vertical angles. True.
PTS: 1
DIF: Average
REF: Lesson 2-3
OBJ: 2-3.4 Write the contrapositive of if-then statements.
NAT: NCTM RP.1
TOP: Write the contrapositive of if-then statements.
KEY: Contrapositive | If-Then Statements
109. ANS:
Sample: Noncongruent segments are not two segments having the same measure. True.
PTS: 1
DIF: Average
REF: Lesson 2-3
OBJ: 2-3.4 Write the contrapositive of if-then statements.
TOP: Write the contrapositive of if-then statements.
110. ANS:
Sample:
Given:
and
are adjacent angles.
Prove: Point Q is in the interior of
.
NAT: NCTM RP.1
KEY: Contrapositive | If-Then Statements
Proof: By the definition of adjacent angles, the two angles have a common side
vertex, Q must be in the interior of
.
S
. Since R is the common
Q
R
T
PTS: 1
DIF: Basic
REF: Lesson 2-5
OBJ: 2-5.2 Write paragraph proofs.
NAT: NCTM RP.1 TOP: Write paragraph proofs.
KEY: Proofs | Paragraph Proofs
111. ANS:
Sample:
Given: Isosceles triangle GHJ;
is a segment between points K and L on line KL; HG = JG.
Prove:
Proof: Since they are linear pairs,
equal angles gives
and
. Since the triangle is isosceles,
. By definition of congruent angles
PTS: 1
DIF: Average
REF: Lesson 2-5
NAT: NCTM RP.1 TOP: Write paragraph proofs.
112. ANS:
Sample:
Given:
Prove:
Proof:
angles,
all intersect at point P;
and
. Substituting,
. Therefore
. Subtracting the
.
OBJ: 2-5.2 Write paragraph proofs.
KEY: Proofs | Paragraph Proofs
and
. If
.
, by definition of congruent
.
PTS: 1
DIF: Basic
REF: Lesson 2-5
NAT: NCTM RP.1 TOP: Write paragraph proofs.
113. ANS:
Sample:
Given: Vertical isosceles triangles;
bisects
;
.
Prove:
OBJ: 2-5.2 Write paragraph proofs.
KEY: Proofs | Paragraph Proofs
Proof:
because vertical angles are congruent.
bisects
, and by definition of a segment
bisector, it bisects at the midpoint. Definition of a midpoint gives two equal segments, which by definition of
congruence leads to
. Since
,
by the AAS Theorem.
PTS: 1
DIF: Basic
REF: Lesson 4-5
OBJ: 4-5.2 Use the AAS Theorem to test for triangle congruence.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Use the AAS Theorem to test for triangle congruence.
KEY: AAS Theorem | Congruent Triangles
114. ANS:
Sample:
Given: Regular hexagon ABCDFG;
each bisect the hexagon and meet at point H;
and
.
Prove:
Proof:
Statements
Reasons
1. Given
1.
and
2.
2. Definition of congruent segments
3.
,
3. Segment Addition Postulate
4.
4. Substitution Property
5.
5. Substitution Property
6.
6. Reflexive Property
7.
7. Subtraction Property
8.
8. Definition of congruent segments
PTS: 1
DIF: Average
REF: Lesson 2-7
OBJ: 2-7.2 Write proofs involving segment congruence.
TOP: Write proofs involving segment congruence.
115. ANS:
Sample:
Given:
.
Prove:
Proof:
Statements
1.
,
,
NAT: NCTM GM.1c | NCTM RP.3
KEY: Proofs | Congruent Segments
;
;
Reasons
1. Given
,
2.
2. Angle Addition Postulate
3.
4.
5.
6.
3. Substitution Property
4. Substitution Property
5. Reflexive Property
6. Subtraction Property
PTS: 1
DIF: Average
REF: Lesson 2-8
OBJ: 2-8.2 Write proofs involving congruent angles.
TOP: Write proofs involving congruent angles.
116. ANS:
Sample:
Given:
Prove:
Proof:
Statements
Reasons
1. Given
1.
NAT: NCTM GM.1c | NCTM RP.3
KEY: Proofs | Congruent Angles
2.
3.
4.
5.
6.
,
are right angles.
,
2. Given
3. Definition of perpendicular
4. Definition of right angle
5. Substitution Property
6. Definition of congruent angles
PTS: 1
DIF: Basic
REF: Lesson 2-8
OBJ: 2-8.3 Write proofs involving right angles.
NAT: NCTM PS.1 | NCTM PS.2 | NCTM PS.3
TOP: Write proofs involving right angles.
KEY: Proofs | Right Angles
117. ANS:
Sample:
Given: Square GHJK
Prove:
Proof:
Statements
Reasons
1. GHJK is a square.
1. Given
2. Definition of a square
2.
3. Definition of a square
3.
4. Reflexive Property
4.
5.
5. SSS Postulate
PTS: 1
DIF: Basic
REF: Lesson 4-4
OBJ: 4-4.1 Use the SSS Postulate to test for triangle congruence.
NAT: NCTM GM.1 | NCTM GM.1b
TOP: Use the SSS Postulate to test for triangle congruence.
KEY: SSS Postulate | Congruent Triangles
118. ANS:
Sample:
Given: W is the midpoint of
and ;
.
Prove:
Proof:
Statements
Reasons
1. Given
1. W is the midpoint of
.
2. Midpoint Theorem
2.
3. Given
3. W is the midpoint of .
4. Midpoint Theoremt
4.
5. Given
5.
6.
6. SSS Postulate
PTS: 1
DIF: Average
REF: Lesson 4-4
OBJ: 4-4.1 Use the SSS Postulate to test for triangle congruence.
NAT: NCTM GM.1 | NCTM GM.1b
TOP: Use the SSS Postulate to test for triangle congruence.
KEY: SSS Postulate | Congruent Triangles
119. ANS:
Sample:
Given: M is the midpoint of
and .
Prove:
Proof:
Statements
1. M is the midpoint of
.
2.
3. M is the midpoint of .
4.
5. The polygon is a regular hexagon.
6.
7.
PTS:
OBJ:
NAT:
KEY:
120. ANS:
Reasons
1. Given
2. Midpoint Theorem
3. Given
4. Midpoint Theorem
5. Given
6. Definition of regular hexagon
7. SSS Postulate
1
DIF: Average
REF: Lesson 4-4
4-4.1 Use the SSS Postulate to test for triangle congruence.
NCTM GM.1 | NCTM GM.1b
TOP: Use the SSS Postulate to test for triangle congruence.
SSS Postulate | Congruent Triangles
M ath Club
33
Theater
27
27
63
Find the number of people who joined the math club only, the number of people who joined theater only, and
the number of people who joined neither math club nor theater. Plot all these calculated values together with
the number of people who joined both on the Venn diagram to represent the data.
PTS: 1
DIF: Average
REF: Lesson 2-2
NAT: NCTM RP.3 TOP: Solve multi-step problems.
121. ANS:
M exico
27
OBJ: 2-2.3 Solve multi-step problems.
KEY: Solve multi-step problems.
Canada
59
47
82
Find the number of people who have been to Mexico only, the number of people who have been to Canada
only, and the number of people who have been to neither Mexico nor Canada. Plot all these calculated values
together with the number of people who have been to both Mexico and Canada on the Venn diagram to
represent the data.
PTS: 1
DIF: Advanced
REF: Lesson 2-2
NAT: NCTM RP.3 TOP: Solve multi-step problems.
OBJ: 2-2.3 Solve multi-step problems.
KEY: Solve multi-step problems.
122. ANS:
Sample answer: If a triangle has three acute angles, then it is an acute triangle; true.
The converse of a conditional statement
exchanges the hypothesis and conclusion of the
conditional. It is also known as
PTS: 1
DIF: Average
REF: Lesson 2-3
OBJ: 2-3.5 Solve multi-step problems.
NAT: NCTM RP.3 | NCTM RP.1
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.
123. ANS:
Sample answer:
Conditional statement: If a figure is a right triangle, then it has one right angle.
Converse: If a triangle has one right angle, then it is a right triangle.
A conditional statement is a statement that can be written in if-then form. The converse of a conditional
statement
exchanges the hypothesis and conclusion of the conditional. It is also known as
PTS: 1
DIF: Average
NAT: NCTM RP.3 | NCTM RP.1
KEY: Solve multi-step problems.
124. ANS:
invalid
REF: Lesson 2-3
OBJ: 2-3.5 Solve multi-step problems.
TOP: Solve multi-step problems.
Law of Detachment is a form of deductive reasoning which is used to draw conclusions from true conditional
statements. In other words, if the conditional statement is true and the hypothesis is true, then according to the
Law of Detachment the conclusion is also true.
PTS: 1
DIF: Average
NAT: NCTM GM.1 | NCTM GM.1a
KEY: Solve multi-step problems.
125. ANS:
REF: Lesson 2-4
OBJ: 2-4.2 Solve multi-step problems.
TOP: Solve multi-step problems.
then Emma goes swimming.
A true conditional statement and a true hypothesis is given in the problem. Use the Law of Detachment to
draw the conclusion.
PTS: 1
DIF: Advanced
REF: Lesson 2-4
OBJ: 2-4.2 Solve multi-step problems.
NAT: NCTM GM.1 | NCTM GM.1a
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.
126. ANS:
He will have 6 different planes and 10 lines.
Points that do not lie on the same line are noncollinear. Points that do not lie in the same plane are
noncoplanar.
PTS: 1
DIF: Average
REF: Lesson 2-5
NAT: NCTM RP.1 TOP: Solve multi-step problems.
127. ANS:
OBJ: 2-5.3 Solve multi-step problems.
KEY: Solve multi-step problems.
If two angles are vertical angles, then they are congruent.
PTS: 1
DIF: Advanced
REF: Lesson 2-8
OBJ: 2-8.4 Solve multi-step problems.
NAT: NCTM GM.1c | NCTM RP.3 | NCTM PS.1 | NCTM PS.2 | NCTM PS.3
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.
128. ANS:
skew lines
Lines that do not intersect and are not coplanar are called skew lines.
PTS: 1
DIF: Advanced
REF: Lesson 3-1
NAT: NCTM GM.1 | NCTM GM.1a | NCTM GM.1b
KEY: Solve multi-step problems.
129. ANS:
3050
OBJ: 3-1.3 Solve multi-step problems.
TOP: Solve multi-step problems.
Calculate the rate of change. Multiply it by 4 and then add the product to the circulation number in 1997 to
predict the circulation for 2001.
PTS: 1
DIF: Average
REF: Lesson 3-3
OBJ: 3-3.3 Solve multi-step problems.
NAT: NCTM AL.2 | NCTM AL.2c | NCTM RE.2 | NCTM GM.1b | NCTM GM.2 | NCTM GM.2a
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.
130. ANS:
Sample answer: 92.5
Find the rate of change. Multiply it by 8 and then add the product to the number of laptop computers used in
1998 to predict the number of laptop computers used in 2006.
PTS: 1
DIF: Advanced
REF: Lesson 3-3
OBJ: 3-3.3 Solve multi-step problems.
NAT: NCTM AL.2 | NCTM AL.2c | NCTM RE.2 | NCTM GM.1b | NCTM GM.2 | NCTM GM.2a
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.
131. ANS:
The slope-intercept form of a linear equation is
where m is the slope of the line and b is the yintercept.
Use the point-slope form and either point to write the equation.
are the coordinates of any point on the line and
is the slope of the
line.
PTS: 1
DIF: Advanced
REF: Lesson 3-4
NAT: NCTM AL.2 | NCTM GM.1 | NCTM GM.1b
KEY: Solve multi-step problems.
OBJ: 3-4.3 Solve multi-step problems.
TOP: Solve multi-step problems.
132. ANS:
An obtuse triangle has one obtuse angle.
PTS: 1
DIF: Average
REF: Lesson 4-1
NAT: NCTM GM.1 | NCTM GM.1a | NCTM GM.1b
KEY: Solve multi-step problems.
133. ANS:
Given: PQRS is a quadrilateral.
Prove:
OBJ: 4-1.3 Solve multi-step problems.
TOP: Solve multi-step problems.
P
S
2
1
4
3
Q
R
Proof:
Statements
1. PQRS is a quadrilateral.
2.
Reasons
1. Given
2. Angle Sum Theorem
3.
4.
5.
3. Addition Property
4. Angle Addition Postulate
5. Substitution
A two-column proof is a deductive argument that contains statements and reasons organized in two columns.
The sum of the measures of the angles of a triangle is 180.
PTS: 1
DIF: Advanced
NAT: NCTM GM.1 | NCTM GM.1b
KEY: Solve multi-step problems.
134. ANS:
Sample:
Given:
is isosceles;
Prove:
Proof:
REF: Lesson 4-1
OBJ: 4-2.3 Solve multi-step problems.
TOP: Solve multi-step problems.
.
PTS: 1
DIF: Average
REF: Lesson 4-5
OBJ: 4-5.1 Use the ASA Postulate to test for triangle congruence.
NAT: NCTM GM.1 | NCTM GM.1b
TOP: Use the ASA Postulate to test for triangle congruence.
KEY: ASA Postulate | Congruent Triangles
135. ANS:
Sample:
Given:
bisects
;
.
Prove:
Proof:
PTS:
OBJ:
NAT:
KEY:
1
DIF: Average
REF: Lesson 4-5
4-5.1 Use the ASA Postulate to test for triangle congruence.
NCTM GM.1 | NCTM GM.1b
TOP: Use the ASA Postulate to test for triangle congruence.
ASA Postulate | Congruent Triangles
136. ANS:
Given:
Prove:
A
D
B
C
Proof: We are given that
so by definition of angle bisector,
By the Reflexive Property,
by ASA.
It is given that
A paragraph proof is a deductive argument that contains statements and reasons organized in a paragraph.
If two angles and the included side of one triangle are congruent to two angles and the included side of
another triangle, then the triangles are congruent.
PTS: 1
DIF: Basic
REF: Lesson 4-5
OBJ: 4-5.3 Solve multi-step problems.
NAT: NCTM GM.1 | NCTM GM.1b | NCTM GM.1a
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.
137. ANS:
Since T is the midpoint of PR,
As
because alternate interior angles are
congruent. Also,
because vertical angles are congruent. So,
by ASA.
If two angles and the included side of one triangle are congruent to two angles and the included side of
another triangle, then the triangles are congruent.
PTS: 1
DIF: Advanced
REF: Lesson 4-5
OBJ: 4-5.3 Solve multi-step problems.
NAT: NCTM GM.1 | NCTM GM.1b | NCTM GM.1a
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.
138. ANS:
Sample:
Given:
is equilateral;.
bisects
.
Prove: T is the midpoint of
.
Proof:
Statements
Reasons
1.
is equilateral.
1. Given
2.
2. Equilateral triangles are equiangular.
3. Definition of equilateral triangle
3.
4. Given
4.
bisects
.
5.
5. Definition of angle bisector
6.
6. ASA Postulate
7. CPCTC
7.
8.
8. Definition of congruence
9. Definition of midpoint
9. T is the midpoint of
PTS: 1
DIF: Basic
REF: Lesson 4-6
OBJ: 4-6.3 Solve multi-step problems.
NAT: NCTM GM.1 | NCTM GM.1a | NCTM GM.2 | NCTM GM.2a
TOP: Use properties of equilateral triangles.
KEY: Equilateral Triangles
139. ANS:
Given:
is equilateral;
.
Prove:
P
Q
R
S
T
Proof:
Statements
1.
2.
3.
4.
5.
6.
7.
Reasons
1. Given
2. Definition of equilateral triangle
3. Given
4. Given
5. SAS
6. CPCTC
7. Isosceles Triangle Theorem
A two-column proof is a deductive argument that contains statements and reasons organized in two columns.
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
PTS: 1
DIF: Average
REF: Lesson 4-6
OBJ: 4-6.3 Solve multi-step problems.
NAT: NCTM GM.1 | NCTM GM.1a | NCTM GM.2 | NCTM GM.2a
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.
140. ANS:
rotation
PTS: 1
DIF: Average
NAT: NCTM GM.2 | NCTM GM.3
KEY: congruence transformation
141. ANS:
Sample:
Given:
with
;
Prove:
Proof:
REF: Lesson 4-7
OBJ: 4-7.3 Solve multi-step problems.
TOP: Congruence transformations.
y
C (b, 2a )
(0.5b, a)
F (1.5 b, a )
D
A (0, 0)
B (2 b, 0)
The slope of CA is
The slope of CD is
The slope of AB is
or
x
. The slope of CB is
or
or
. The slope of CF is
or . The slope of DF is
or
.
or .
because they are formed by segments of equal slopes.
because they are formed by segments of equal slopes.
because they are the same angle.
.
Therefore,
PTS: 1
DIF: Average
REF: Lesson 4-8
NAT: NCTM PS.1 | NCTM PS.2 | NCTM PS.3
KEY: Proofs | Coordinate Proofs
142. ANS:
Sample:
Given:
with coordinates
,
,
Prove:
Proof:
.
OBJ: 4-8.2 Solve multi-step problems.
TOP: Write coordinate proofs.
is isosceles
y
C (a, b)
A (0, 0)
Since
B (2 a, 0)
,
x
and by definition the triangle is isosceles.
PTS: 1
DIF: Basic
REF: Lesson 4-8
NAT: NCTM PS.1 | NCTM PS.2 | NCTM PS.3
OBJ: 4-8.2 Solve multi-step problems.
TOP: Write coordinate proofs.
KEY: Proofs | Coordinate Proofs
143. ANS:
Sample:
Given:
with
;
Prove:
Proof:
y
C (0, 2 a)
D ( a, a)
A (0, 0)
Therefore,
B (2 a, 0)
x
by SSS Theorem.
PTS: 1
DIF: Basic
REF: Lesson 4-8
NAT: NCTM PS.1 | NCTM PS.2 | NCTM PS.3
KEY: Proofs | Coordinate Proofs
144. ANS:
OBJ: 4-8.2 Solve multi-step problems.
TOP: Write coordinate proofs.
If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater
measure than the angle opposite the shorter side.
PTS: 1
DIF: Average
REF: Lesson 5-3
OBJ: 5-3.3 Solve multi-step problems.
NAT: NCTM GM.1 | NCTM GM.1a | NCTM AL.2 | NCTM AL.2b | NCTM RP.1
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.
145. ANS:
If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater
measure than the angle opposite the shorter side.
PTS: 1
DIF: Advanced
REF: Lesson 5-3
OBJ: 5-3.3 Solve multi-step problems.
NAT: NCTM GM.1 | NCTM GM.1a | NCTM AL.2 | NCTM AL.2b | NCTM RP.1
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.
146. ANS:
Sample:
Given:
Prove: is positive.
Indirect Proof:
Step 1: Assume a is not positive, or
Step 2:
Case 1:
.
Case 2:
Step 3: Case 1 contradicts a being positive and Case 2 is a contradiction since no number is smaller than
itself. Hence, the assumption must be false, and a is positive.
PTS: 1
DIF: Average
REF: Lesson 5-4
OBJ: 5-4.1 Use indirect proof with algebra.
NAT: NCTM GM.1c | NCTM RP.1
TOP: Use indirect proof with algebra.
KEY: Proofs | Indirect Proofs
147. ANS:
Sample:
Given:
,
Prove:
Indirect Proof:
Step 1: Assume
.
Step 2:
(Angle Sum Theorem)
(Subtraction Property)
because
.
(Linear Pair)
because
.
Step 3: This contradicts the given fact
. Thus,
.
PTS: 1
DIF: Basic
REF: Lesson 5-4
OBJ: 5-4.2 Use indirect proof with geometry.
NAT: NCTM AL.2 | NCTM AL.2b | NCTM GM.1
TOP: Use indirect proof with geometry.
KEY: Proofs | Indirect Proofs
148. ANS:
Sample:
Given:
; A is the midpoint of BF and CD.
Prove:
Indirect Proof:
Step 1: Assume
is not parallel to
.
Step 2: If
is not parallel to
, then
is not congruent to
.
If
is not parallel to
, then
is not congruent to
.
If those angles are not congruent, then the triangles cannot be congruent.
However,
by the midpoint theorem, and
because they are
vertical angles.
by SAS.
and
because corresponding
parts of congruent triangles are congruent.
Step 3: The assumption is false, so
.
PTS: 1
DIF: Average
REF: Lesson 5-4
OBJ: 5-4.2 Use indirect proof with geometry.
NAT: NCTM AL.2 | NCTM AL.2b | NCTM GM.1
TOP: Use indirect proof with geometry.
KEY: Proofs | Indirect Proofs
149. ANS:
Sample:
Given:
with angle measures as shown
Prove:
Indirect Proof:
Step 1: Assume
.
Step 2: By angle-side relationships
.
Step 3: This contradicts the given angle measures, so the assumption must be false and
.
PTS: 1
DIF: Average
REF: Lesson 5-4
OBJ: 5-4.2 Use indirect proof with geometry.
NAT: NCTM AL.2 | NCTM AL.2b | NCTM GM.1
TOP: Use indirect proof with geometry.
KEY: Proofs | Indirect Proofs
150. ANS:
Yes, if you assume that Megan was the winner of the race, then she must have taken less time than the time
taken by any of the participants in the race. But this contradicts the fact that Megan took more time than Sara
to complete the race. Thus, the assumption that Megan was the winner of the race is false.
Assume that the conclusion is false.
Show that this assumption leads to a contradiction of the hypothesis, or some other fact, such as a definition,
postulate, theorem, or corollary.
Point out that because the false conclusion leads to an incorrect statement, the original conclusion must be
true.
PTS: 1
DIF: Advanced
REF: Lesson 5-4
OBJ: 5-4.3 Solve multi-step problems.
NAT: NCTM GM.1c | NCTM RP.1 | NCTM AL.2 | NCTM AL.2b | NCTM GM.1
TOP: Solve multi-step problems.
KEY: Solve multi-step problems.