Pre-Test Unit 2: Cong. and Similarity KEY
No calculator necessary. Please do not use a calculator.
Answers the following questions. (5 pts; no partial credit)
1. List all the transformations that lead to congruent images.
Translation, rotation, reflection
2. List all the transformations that lead to similar, but not congruent, images.
Dilation
Apply the given transformation to the given pre-image. (5 pts; no partial credit)
3. Translation by vector 4. Rotation by 90°
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Apply the given transformation to the given pre-image. (5 pts; no partial credit)
5. Reflection across the -axis
6. Dilation by scale factor 8
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Apply the given series of transformations to the given pre-image. (5 pts; 2 pts for each transformation, 1 pt for
correct order)
7. Translation by vector and rotation by 180°
8. Dilation by scale factor 2 and reflect across -axis
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Identify a specific series of transformations that would take the pre-image (darker in blue) to the image (lighter
in green). Then tell whether the pre-image and image are congruent or similar.
(5 pts; 2 pts for transformation(s), 2 pts for specific vector(s), rotation angle(s), reflection line(s), or scale factor(s),
1 pt for cong/sim)
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Dilate by c = and similar
Reflect across y-axis and congruent
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Dilate by c = and reflect across y-axis
Similar
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Rotate 90° and translate by Or translate by and rotate by 90°; Congruent
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Find the angle measure of each missing angle. (5 pts; 3 pts for computation error or only solving for the variable
on problem 13)
F
13.
14.
F
2w°
75°
g°
90°
w°
G
35°
G
H
H
= 30° and 2 = 60°
= 110°
Determine if the following triangles are similar or not and explain why or why not. (5 pts; 2 pts for answer, 3 pts
for explanation)
15.
B
75°
Similar because all angle measures are congruent: 20°, 75°, 85°
20°
A
C
B'
85°
20°
A'
C'
Use the picture to answer the following questions. (5 pts; no partial credit)
16. Name a pair of corresponding
angles.
1 ≅ 3, 2 ≅ 4
5 ≅ 7, or 6 ≅ 8
BC DE
B
∠1
∠2
17. Name a pair of alternate
interior angles.
∠5
∠6
C
2≅
7 or
6≅
3
18. Name a pair of alternate
exterior angles.
∠3
D ∠4
5≅
4 or
1≅
8
∠7
19. Name a pair of vertical angles.
1 ≅ 6, 2 ≅ 5
3 ≅ 8, or 4 ≅ 7
∠8
E
20. If ∠1 = 135°, what is ∠7?
7 = 45°
4
Unit 2 Homework Key
Lesson 2.1
Perform the given dilation on each given pre-image.
1. Dilate by = , center (0,0)
2. Dilate by = , center (2,2)
3. Dilate by = , center (0,0)
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4. Dilate by = 2, center (6,4)
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5. Dilate by = , center (0,0)
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6. Dilate by = , center (−6,2)
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7. Dilate by = , center (0,0)
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8. Dilate by = , center (−3, −6)
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9. Dilate by = , center (0,0)
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10. Dilate by = , center (4,4)
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11. Dilate by = , center (0,0)
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13. Dilate by = 2, center (0,0)
14. Dilate by = , center (−4, −2)
15. Dilate by = , center (0,0)
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16. Dilate by = , center (3,0)
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17. Dilate by = , center (0,0)
18. Dilate by = , center (0, −6)
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12. Dilate by = , center (−4,8)
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19. Dilate the triangle by a scale factor = using the
origin as the center of dilation. Which of the following
statements will be true? For each explain why or why not.
a. Point $′ is at the same coordinates as Point $.
No, it would only be same if it were the center of dilation.
b. Point &′ is at the same coordinates as Point &.
No, it would only be same if it were the center of dilation.
c. Point '′ is at the same coordinates as Point '.
No, it would only be same if it were the center of dilation.
d. The image’s perimeter equals the pre-image’s.
No, the perimeter is half.
e. The image’s area equals the pre-image’s.
No, the area is a fourth since it’s half on each side.
f. The image is congruent to pre-image.
No, it is similar.
((((( is horizontal.
g. Line segment $′'′
Yes, it shrunk but did not rotate.
20. Dilate the triangle by a scale factor = 2 using the
point !4,4" as the center of dilation. Which of the following
statements will be true? For each explain why or why not.
a. Point $′ will be at the same coordinates as Point $.
No, it would only be same if it were the center of dilation.
b. Point &′ will be at the same coordinates as Point &.
No, it would only be same if it were the center of dilation.
c. Point '′ will be at the same coordinates as Point '.
Yes, it is the center of dilation and doesn’t move.
d. The image’s perimeter equals the pre-image’s.
No, the perimeter is double.
e. The image’s area equals the pre-image’s.
No, the area is quadruple since each side doubles.
f. The image is similar to pre-image.
Yes, it is the same shape but different size.
(((((( is vertical.
g. Line segment &′'′
Yes, it enlarged but did not rotate.
21. In your own words, explain what a dilation does to a pre-image. Remember to consider the center of dilation
in your explanation.
A dilation shrinks everything towards the center of dilation or enlarges everything away from the center of
dilation by the given scale factor.
22. How could dilations be used in real life?
Answers will vary
7
Lesson 2.2
Perform the given reflection or series of transformations on each given pre-image.
1. Reflect across -axis
2. Reflect across -axis
3. Reflect across -axis
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4. Reflect across -axis
5. Reflect across -axis
6. Reflect across -axis
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7. Reflect across = −2
8. Reflect across #2
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9. Reflect across 10. Reflect across -axis
and dilate by =
11. Reflect across -axis
and dilate by =
12. Reflect across -axis
and dilate by =
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13. Dilate by = 2
and reflect across -axis
14. Dilate by =
and reflect across -axis
15. Dilate by =
and reflect across -axis
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16. Reflect across -axis
and dilate by =
17. Reflect across -axis
and dilate by = 18. Reflect across -axis
and dilate by = 8
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19. Reflect the triangle across the line = 2. Which of the
following statements will be true? For each explain why or
why not.
a. Point $′ is at the same coordinates as Point $.
Yes, because it is one the line of reflection.
b. Point &′ is at the same coordinates as Point &.
No, because it is not on the line of reflection.
c. Point '′ is at the same coordinates as Point '.
No, because it is not on the line of reflection.
d. The image’s perimeter equals the pre-image’s.
Yes, they are congruent shapes.
e. The image’s area equals the pre-image’s.
Yes, they are congruent shapes.
f. The image is congruent to pre-image.
Yes, it is the same size and shape.
((((( is horizontal.
g. Line segment $′'′
Yes, since it flipped left and right it is still horizontal.
20. Reflect the triangle across the line . Which of the
following statements will be true? For each explain why or
why not.
a. Point $′ will be at the same coordinates as Point $.
No, because it is not on the line of reflection.
b. Point &′ will be at the same coordinates as Point &.
No, because it is not on the line of reflection.
c. Point '′ will be at the same coordinates as Point '.
Yes, because it is on the line of reflection.
d. The image’s perimeter equals the pre-image’s.
Yes, they are congruent shapes.
e. The image’s area equals the pre-image’s.
Yes, they are congruent shapes.
f. The image is similar to pre-image.
No, they are congruent shapes.
g. Line segment ((((((
&′'′ is vertical.
No, since the line of reflection was not vertical or horizontal
21. In your own words, explain what a reflection does to a pre-image. Remember to consider the line of
reflection in your explanation.
It flips a the pre-image across the line of reflection so that it is the same distance from the line but in the opposite
direction.
22. How could reflections be used in real life?
Answers will vary.
10
Lesson 2.3
Perform the given rotation or series of transformations on each given pre-image. When performing a dilation or rotation,
use the origin as the center of dilation or rotation unless it is specified otherwise.
1. Rotate 90°, center: !0,0"
2. Rotate 180°, center: !0,0"
3. Rotate 270°, center: !0,0"
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4. Rotate 270°, center: !0,0"
5. Rotate 180°, center: (1,3)
6. Rotate 90°, center: !2,2"
8. Rotate 90°, center: !0,0"
9. Rotate 180°, center: !0,4"
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7. Rotate 270°, center: !4,0"
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10. Rotate 90°
and reflect across -axis
11. Rotate 180°
and dilate by =
12. Rotate 270°
and reflect across -axis
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13. Dilate by = 2
and reflect across -axis
14. Dilate by = and rotate by 90°
15. Rotate 180°
and reflect across -axis
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16. Reflect across -axis
and rotate 90°
17. Dilate by = and reflect across -axis
18. Dilate by = 2
and rotate 90°
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19. Rotate the triangle 90° around the center of rotation
!3,2". Which of the following statements will be true? For
each explain why or why not.
a. Point $′ is at the same coordinates as Point $.
No, it is not the center of rotation.
b. Point &′ is at the same coordinates as Point &.
No, it is not the center of rotation.
c. Point '′ is at the same coordinates as Point '.
No, it is not the center of rotation.
d. The image’s perimeter equals the pre-image’s.
Yes, the shapes are congruent.
e. The image’s area equals the pre-image’s.
Yes, the shapes are congruent.
f. The image is congruent to pre-image.
Yes, the shapes are the same size and shape.
g. Line segment (((((
$′'′ is horizontal.
No, it only rotated a quarter turn.
20. Rotate the triangle 180° around the center of rotation
!4,0". Which of the following statements will be true? For
each explain why or why not.
a. Point $′ will be at the same coordinates as Point $.
No, it is not the center of rotation.
b. Point &′ will be at the same coordinates as Point &.
Yes, it is the center of rotation.
c. Point '′ will be at the same coordinates as Point '.
No, it is not the center of rotation.
d. The image’s perimeter equals the pre-image’s.
Yes, the shapes are congruent.
e. The image’s area equals the pre-image’s.
Yes, the shapes are congruent.
f. The image is similar to pre-image.
No, the shapes are congruent.
(((((( is vertical.
g. Line segment &′'′
Yes, because it was a half turn so the orientation is still
vertical.
21. In your own words, explain what a rotation does to a pre-image. Remember to consider the center of
rotation in your explanation.
A rotation turns the pre-image about the center of rotation the specified number of degrees.
22. How could rotations be used in real life?
Answers will vary.
13
Lesson 2.4
Perform the given translation or series of transformations on each given pre-image. When performing a dilation or
rotation, use the origin as the center of dilation or rotation.
1. Translate by )
*
*
2. Translate by )
3. Translate by 8
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4. Translate by *
5. Translate by *
6. Translate by 8
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7. Translate by 8. Translate by 8
)
9. Translate by 8
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10. Rotate 90°
and translate by 11. Rotate 180°
and translate by )
12. Reflect across -axis
and translate by )
)
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13. Reflect across -axis
and translate
14. Dilate by = *
by and translate
15. Dilate by = 2
by and translate by *
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16. Dilate by = , rotate 90°,
and reflect across -axis
17. Dilate by = , rotate 180°,
and translate
by 18. Dilate by = 2, reflect across
-axis, and translate by 8
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19. Translate the triangle by vector . Which of the
following statements will be true? For each explain why or
why not.
a. Point $′ is at the same coordinates as Point $.
No, only a zero vector would not change coordinates.
b. Point &′ is at the same coordinates as Point &.
No, only a zero vector would not change coordinates.
c. Point '′ is at the same coordinates as Point '.
No, only a zero vector would not change coordinates.
d. The image’s perimeter equals the pre-image’s.
Yes, the shapes are congruent.
e. The image’s area equals the pre-image’s.
Yes, the shapes are congruent.
f. The image is congruent to pre-image.
Yes, the shapes are the same size and shape.
g. Line segment (((((
$′'′ is horizontal.
Yes, translations don’t change orientation.
20. Translate the triangle by vector *
. Which of the
following statements will be true? For each explain why or
why not.
a. Point $′ will be at the same coordinates as Point $.
No, only a zero vector would not change coordinates.
b. Point &′ will be at the same coordinates as Point &.
No, only a zero vector would not change coordinates.
c. Point '′ will be at the same coordinates as Point '.
No, only a zero vector would not change coordinates.
d. The image’s perimeter equals the pre-image’s.
Yes, the shapes are congruent.
e. The image’s area equals the pre-image’s.
Yes, the shapes are congruent.
f. The image is similar to pre-image.
No, the shapes are congruent.
(((((( is vertical.
g. Line segment &′'′
Yes, translations don’t change orientation.
21. In your own words, explain what a translation does to a pre-image. Remember to consider the translation
vector in your explanation.
A translation takes a shape and slides it horizontally and vertically based on the translation vector.
22. How could translations be used in real life?
Answers will vary.
16
Lesson 2.5
Determine the specific series of transformations that took the pre-image (darker in blue) to the image (lighter in green). Be
sure to give the specific vector, rotation angle, line of reflection and/or scale factor. Then determine if the pre-image and
image are similar or congruent.
1. Rotate 90°, Reflect -axis; Con
+
2. Reflect -axis, Translate ; Con
3. Rotate 90°, Translate ; Con
+
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4. Rotate 180°, Translate ; Con
5. Dilate = 2, Translate )
; Sim
6. Rotate 90°, Reflect -axis; Con
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7. Rotate 270°; Con
9. Reflect across -axis; Con
8. Dilate ; Sim
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10. Dilate = , Rotate 180°,
11. Rotate 270°, Reflect -axis; Con
Translate ; Sim
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,
12. Rotate 90°, Translate +
; Con
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13. Rotate 90°, Translate ; Con
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14. Dilate = , Reflect -axis; Sim
15. Reflect -axis, Translate )*; Con
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16. Rotate 90°, Translate ; Con
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17. Reflect -axis, Translate )
; Con
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18. Rotate 270°, Translate ; Con
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Lesson 2.6
Solve for the variable.
1.
2.
°
3.
15°
100°
10°
°
138°
25°
25°
x = 55
7°
y = 140
4.
z = 32
5.
6.
8°
38°
29°
28°
a = 30
c = 70
8.
9.
32°
72° °
g = 108
63°
<° (< + 56)°
ℎ°
h = 95
10.
( − 10)°
°
(9 + 16)°
b = 41
7.
j = 62
11.
12.
42°
28°
=°
88°
d = 105
( − 20)°
9°
(? + 12)°
?°
>°
147°
(? + 60)°
e = 116
19
f = 36
13.
14.
°
15.
25°
110°
17°
°
118°
25°
35°
x = 45
7°
y = 120
16.
z = 45
17.
18.
81°
40°
9°
8°
29°
28°
a = 33
b = 39
19.
c = 80
20.
21.
39°
64° °
g = 116
61°
<° (< + 50)°
ℎ°
h = 100
22.
j = 65
23.
24.
52°
37°
=°
76°
d = 85
( − 20)°
°
(9 + 24)°
(? + 18)°
?°
>°
137°
(? + 42)°
e = 113
20
f = 40
Lesson 2.7
Decide if the following triangles are similar and explain why using the angle-angle criterion.
1. Triangle 1 – 1 45°, 2 45°
Triangle 2 – 1 45°, 2 90°
Similar
2. Triangle 1 – 1 75°, 2 65°
Triangle 2 – 1 65°, 2 140°
Not similar, #2 not even a triangle
3. Triangle 1 – 1 50°, 2 30°
Triangle 2 – 1 30°, 2 100°
Similar
4. Triangle 1 – 1 80°, 2 20°
Triangle 2 – 1 80°, 2 80°
Similar
5. Triangle 1 – 1 60°, 2 20°
Triangle 2 – 1 40°, 2 100°
Not similar
6. Triangle 1 – 1 45°, 2 30°
Triangle 2 – 1 30°, 2 100°
Not similar
7. Triangle 1 – 1 40°, 2 30°
Triangle 2 – 1 90°, 2 30°
Not similar
8. Triangle 1 – 1 80°, 2 40°
Triangle 2 – 1 40°, 2 60°
Similar
9. Triangle 1 – 1 35°, 2 95°
Triangle 2 – 1 35°, 2 40°
Not similar
10. Triangle 1 – 1 105°, 2 35°
Triangle 2 – 1 40°, 2 105°
Similar
11. Triangle 1 – 1 35°, 2 95°
Triangle 2 – 1 35°, 2 50°
Similar
12. Triangle 1 – 1 50°, 2 50°
Triangle 2 – 1 50°, 2 90°
Not similar
13. Triangle 1 – 1 25°, 2 115°
Triangle 2 – 1 25°, 2 40°
Similar
14. Triangle 1 – 1 70°, 2 45°
Triangle 2 – 1 45°, 2 65°
Similar
15. Triangle 1 – 1 5°, 2 15°
Triangle 2 – 1 120°, 2 15°
Not similar
16. Triangle 1 – 1 90°, 2 20°
Triangle 2 – 1 90°, 2 80°
Not similar
17. Triangle 1 – 1 5°, 2 15°
Triangle 2 – 1 160°, 2 15°
Similar
18. Triangle 1 – 1 80°, 2 30°
Triangle 2 – 1 70°, 2 30°
Similar
19. Triangle 1 – 1 45°, 2 55°
Triangle 2 – 1 55°, 2 90°
Not similar
20. Triangle 1 – 1 72°, 2 23°
Triangle 2 – 1 85°, 2 23°
Similar
21
Lesson 2.8
Use the following picture to answer the questions.
1
5
2
6
3
7
4
8
1. Name a pair of vertical angles.
1≅ 6
2≅ 5
3≅ 8
4≅ 7
2. Name a pair of corresponding angles.
1≅ 3
2≅ 4
5≅ 7
6≅ 8
3. Name a pair of alternate interior angles.
2≅ 7
6≅ 3
4. Name a pair of alternate exterior angles.
1≅ 8
5≅ 4
5. If 2 110°, what is 5?
5 110°
6. If 2 110°, what is 4?
4 110°
7. If 2 110°, what is 7?
7 110°
8. If 1 70°, what is 8?
8 70°
9. If 1 70°, what is 7?
7 110°
10. If 2 140°, what are the measures of all the other angles?
1 40°
3 40°
4 140° 5 140°
6 40°
7 140° 8 40°
22
1
2
3
5
6
4
7
8
11. Name all the pairs of vertical angles.
1≅ 3
2≅ 4
6≅ 8
5≅ 7
12. Name all the pairs of corresponding angles.
1≅ 5
2≅ 6
3≅ 7
4≅ 8
13. Name all the pairs of alternate interior angles.
1≅ 7
4≅ 6
14. Name all the pairs of alternate exterior angles.
2≅ 8
3≅ 5
15. If 2 40°, what is 8?
8 40°
16. If 2 40°, what is 4?
4 40°
17. If 1 140°, what is 7?
7 140°
18. If 1 140°, what is 5?
5 140°
19. If 1 140°, what is 6?
6 40°
20. If 2 35°, what are the measures of all the other angles?
1 145° 3 145° 4 35°
5 145°
6 35°
7 145° 8 35°
23
Review Unit 2: Cong. and Similarity KEY
You may use a calculator.
Unit 2 Goals
• Verify experimentally the properties of rotations, reflections, and translations: a) Lines are taken to lines, and
line segments to line segments of the same length; b) angles are taken to angles of the same measure; c)
parallel lines are taken to parallel lines. (8.G.1)
• Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first
by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that
exhibits the congruence between them. (8.G.2)
• Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using
coordinates. (8.G.3)
• Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by
a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures,
describe a sequence that exhibits the similarity between them. (8.G.4)
• Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the
angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of
triangles. (8.G.5)
Answers the following questions.
3. If you translate the pre-image A, will the image A’ be congruent or only similar? Congruent
4. If you dilate the pre-image B, will the image B’ be congruent or only similar? Similar
5. If you rotate the pre-image C, will the image C’ be congruent or only similar? Congruent
6. If you reflect the pre-image D, will the image D’ be congruent or only similar? Congruent
Name the specific transformation shown in each picture as a translation, rotation, reflection, or dilation. Then
determine if the pre-image (darker in blue) and the image (lighter in green) are similar or congruent.
5. Translation by +
6. Reflection across the x-axis
8
8
6
6
4
4
2
2
5
5
5
5
2
2
4
4
6
6
8
8
24
Apply the given transformation or series of transformations to the given pre-image.
7. Translation by vector +
8. Rotation by 180°
8
8
6
6
4
4
2
2
5
5
5
5
2
2
4
4
6
6
8
8
9. Reflection across the -axis
10. Dilation by scale factor 8
8
6
6
4
4
2
2
5
5
5
5
2
2
4
4
6
6
8
8
11. Translation by vector and rotation by 90°
12. Rotation by 180° and reflect across -axis
8
8
6
6
4
4
2
2
5
5
5
5
2
2
4
4
6
6
8
8
25
Identify a specific series of transformations that would take the pre-image (darker in blue) to the image (lighter
in green). Then tell whether the pre-image and image are congruent or similar.
13. Reflect across -axis, translate by vector +
,
14. Rotate 90°, reflect across the -axis
8
8
6
6
4
4
2
2
5
5
5
5
2
2
4
4
6
6
8
8
15. Dilate by scale factor 2, translate by vector +
,
16. Rotate 180°, translate by vector )*
8
8
6
6
4
4
2
2
5
5
5
5
2
2
4
4
6
6
8
8
Find the angle measure of each missing angle.
17. = = 140°
18. > = 130°
A
A
80°
d°
B
e°
40°
B
C
26
50°
C
19. = 50°, ( + 10) = 60°
20. 2 = 40°, 3 = 60°, 4 = 80°
D
D
70°
E
4x°
(x+10)°
x°
E
F
3x°
2x°
F
Determine if the following triangles are similar or not and explain why or why not.
21. Similar, same angles
22. Not similar, not same angles
G'
H
110°
110°
G
23. Not similar, not same angles
H
H
110°
I
40°
30°
G
I'
G'
I
50°
30°
I'
G
I
G'
I'
130°
H'
H'
H'
Use the picture to answer the following questions.
24. What type of angles are ∠1and∠4?
Vertical angles
K
∠ 1 ∠2
∠3 ∠ 4
KL MN
25. What type of angles are ∠1and∠5?
Corresponding angles
26. What type of angles are ∠1and∠8?
Alternate exterior angles
L
27. What type of angles are ∠3and∠6?
Alternate interior angles
M
∠ 5 ∠6
∠ 7 ∠8
28. List all the angles congruent to ∠4.
∠1,∠5,∠8
29. If ∠1 = 75°, what is ∠8?
∠8 = 75°
N
30. If ∠2 = 135°, what is ∠4?
∠4 = 45°
27