IT N U
2
Review
Prove the following. 1.
Prove that vertical angles 1 and 3 are congruent.
2.
1 4 2 3
___
In nABC below, is a perpendicular ___ CD bisector of AB . Prove that nADC nBDC using a triangle congruence postulate or theorem. Then, describe a transformation that could carry nADC onto nBDC. C
A
D
B
Determine if the figures are similar. If they are similar, describe a dilation (including the scale factor) that would transform one figure into the other. If not, explain how you know. E W 5
3
104
Z
7.5
100°
140°
65°
X
7 55°
9
H Y
4.5 F 140°
4. 100°
O
14 53°
10.5 8
65° 13.5
10
55° G
Unit 2 Review
N
M
127° 8 P
S
6 53°
R
4 5 127° T 3 Q
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3.
Choose the best answer. 5.
___
midpoint of AB , and E is In nABC, D is the ___ the midpoint of AC . What must be true of this figure? A. B.
6.
Which pair of figures must be similar? A. a pair of equilateral triangles B. a pair of isosceles triangles
___ DE is perpendicular to AB . ___
C. a pair of obtuse triangles
___ is parallel to BC DE . ___
D. a pair of right triangles
DE AD C. ___ BC 5 ___ AC
D. DE 5 AB ____
___
7. MQ and NR intersect at point P. Can it be shown that nMNP , nQRP? M
Using only the information given, can it be shown that nCDE , nFGE? C
12 P N 10
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8.
15
18
R Q
20
10 D
15
F
12 E
9
6 G
A. yes, by the AA , Postulate
A. yes, by the AA , Postulate
B. yes, by the SAS , Theorem
B. yes, by the SAS , Theorem
C. yes, by the SSS , Theorem
C. yes, by the SSS , Theorem
D. No, the information given is not sufficient.
D. No, the information given is not sufficient.
Dilate each line or segment as indicated. State whether the resulting image is parallel to the preimage, collinear with the preimage, or neither parallel nor collinear with the preimage.
( 2
2
)
9. D__2(x, y) 5 __ 3 x, __ 3 y 3
10. Scale factor: 2 Center of dilation: (4, 3)
y 6
y
(3, 6)
5
6
4
5
3
4
2
3
1 0
(0, 0) 1 2 3 4 5 6
x
2
(5, 4) (2, 3)
1 0
1 2 3 4 5 6 7 8
x
Unit 2 Review
105
Use a compass and straightedge for questions 11–13. ___›
11. Construct the bisector of /LMN. Label the bisector MP .
N
L
M
‹___›
‹___›
12. Construct a line perpendicular to AB that passes through point C. Label it CD .
C
13. Inscribe a square in the circle below.
106
Unit 2 Review
B
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A
Choose the best answer. 15. Which pair of rigid motions could be used to show that figure 1 is congruent to figure 2?
14. Which pair of rigid motions could be used to show that nABC and nA9B9C9 are congruent? y B
6
Figure 1
4
A –6
C C –4
0
–2
Figure 2
2 2
4
x
6
–2 –4
B
A
–6
A. horizontal reflection of Figure 1 followed by a translation to the right
A. reflection of nABC across the x-axis followed by a translation of 1 unit left
B. vertical reflection of Figure 1 followed by a translation to the right
B. reflection of nABC across the x-axis followed by a translation of 1 unit up
C. rotation of Figure 1 90° clockwise followed by a translation to the right
C. rotation of nABC 90° counterclockwise about the origin followed by a translation of 1 unit left
D. rotation of Figure 1 90° counterclockwise followed by a translation to the left
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D. rotation of nABC 90° counterclockwise about the origin followed by a translation of 1 unit up
Solve. 16. Describe how nDEF was transformed to its image, nD9E9F9, both in words and in function notation. y
Words:
6 4 2
D –4
–2
D
0
–2
E E F 2
4
F
6
8
x
Function Notation:
–4 –6
Unit 2 Review
107
Fill in the blanks to complete the proof. PROVE
The medians of a triangle meet at a single point. ___
___
In nABC, E is the midpoint of AB , and D is the midpoint of BC . ___
___
17. In nABC, EC and AD are medians that intersect at point P. AE 5 BE and CD 5 BD
18. A line is added to connect points E and D. ‹__›
___
because medians extend from a vertex to the
of the opposite side.
to AC and ED must equal
? AC.
B
B
E
E
D
D
P
P
A
A
C ‹___›
and AC 19. ED are
transversal
‹__› EC .
lines cut by Angles /DEC and /ACE
are
C
20. /EPD and /CPA are angles, so /EPD /CPA. Triangles PED and PCA are similar according to
angles, so /DEC /ACE.
. B
B E
D
E
P A
P C
ED
EP
21. ___ AC 5
so ___ CP 5
D
A DP
and ___ AP 5
C 2
. So, point P is __ 3 of the way from point C to
2
point E and __ 3 of the way from point A to point D. Any two medians could have been chosen for this proof, so this is true for all three medians. All three medians meet at the single point 2
that is __ 3 of the way down each median.
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Unit 2 Review
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‹__›
___ ___
___
Since ED bisects AB and BC , ED must be
Solve.
A 4
22. What postulate or theorem can be used to show that nABC , nAB9C9? Cite the relevant angles and/or sides. Then describe a dilation that would transform nABC , nAB9C9.
C
B
8
C
23.
EXTEND Triangle___ RST is shown. Sarah reflects only side RT and angles RTS and TRS across the y-axis. How many different triangles can be formed if she extends the rays of her reflected image until they intersect? Which congruence postulate does this illustrate?
B
24. PROVE Prove that opposite angles of a parallelogram are congruent. Draw on the figure below to help explain your reasoning. B
C
y
A
6
T
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–6
T
4
R 2 –4
–2
0
D
R
S 2
4
6
x
–2 –4 –6
Unit 2 Review
109
Performance Task
For this activity, you will work in pairs or small groups to help design a community park. Some features of the park are already represented on the diagram below. Use your compass, straightedge, and knowledge of constructions to add the features described in the steps on the next page.
Community Park Benches A and B
N W
E S
Fountain A
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Fountain B
110
Unit 2 Performance Task
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1.
The landscaper wants to design a set of right triangular benches. Bench A and Bench B are shown on the diagram. He wants there to be 4 benches, arranged like petals of a flower. Draw Bench C, which is a rotation of Bench A 180 about its southern vertex (the point it shares with Bench B). Draw Bench D by rotating Bench B 180 about the same point.
2.
Are the benches congruent figures? How do you know?
3.
The landscaper wants to set aside two rectangular patches of land for fountains. The first, Fountain A, is shown. Fountain B will have sides that are twice as long as the corresponding sides of Fountain A. Construct Fountain B on the diagram, placing its northwest corner on the point shown. (Hint: Recall how to construct a line through a point and parallel to another line.)
4.
Describe a transformation that could be used to produce the model of Fountain B from the model of Fountain A. Are Fountain A and Fountain B similar figures? How do you know?
5.
Next, the landscaper wants to place a mosaic in the northeast portion of the park. The design on the mosaic will feature a regular hexagon inscribed in a circle. Construct an inscribed regular hexagon on the diagram to represent the mosaic.
6.
Last, the landscaper wants to construct two identical triangular flowerbeds from wooden planks. He has placed three wooden planks—one 5-meter plank, one 12-meter plank, and one 13-meter plank—by each fountain. He begins to assemble the flowerbed near Fountain A and sends you to assemble the other flowerbed, but does not tell you how to arrange the sides of the triangle. Will the two flowerbeds be congruent? How do you know?
Unit 2 Performance Task
111
