LESSON
Chapter Review
4 4-1 Classifying Triangles
W
Classify each triangle by its angle measure. 1. XYZ
2. XYW
Acute
Z
3. XZW
Equiangular
25
Obtuse
35
X
Classify each triangle by its side lengths. 4. DEF
60
2
5. DEG
D
G
6. EFG
8
Equilateral
Scalene
Y
Isosceles
10
E
F
4-2 Angle Relationships in Triangles Find each angle measure. D
7. mACB 85°
4x – 45
A
8. mK C
2x – 3
K
47° x–7
25° 3x 2x N
M B
9. A carpenter built a triangular support structure for a roof. Two of the angles of the structure measure 32.5° and 47.5°. Find the measure of the third angle.
L
100°
4-3 Congruent Triangles Given ABC XYZ. Identify the congruent corresponding parts. 10. BC
YZ
11. ZX
12. A
X
13. Y L
Given JKL PQR. Find each value. 14. x
Copyright © by Holt, Rinehart and Winston. All rights reserved.
B
J
15
93
R
3y – 2 42°
15. RP 14
CA
K 2y – 1
22 3x°
Q
P
Geometry
CHAPTER 4 REVIEW CONTINUED
CD ; AB AC ; AD ⊥ CB ; AD ⊥ XW ; XAC WAB 16. Given: k; BD Prove: ABD ACD
ᐉ
x
k
y
z C
Statements
w
A
D
B
Reasons
CD ; AB AC ; 1. BD
1. Given
AD 2. AD
2. Reflexive Property of Congruence
; AD ⊥ XW ⊥ CB 3. k; AD
3. Given
4. ADB and ADC are right angles.
4. Def. of ⊥ lines
5. ADB ADC
5. Rt. Thm
6. XAC WAB
6. Given
7. XAC ACD; WAB ABD
7. Parallel lines cut by a transversal, alternate interior angles are congruent.
8. ACD ABD
8. Transitive Property of Congruence
9. CAD BAD
9. Third Angles Theorem
10. ABD ACD
10. Def of Congruent Triangles
4-4 Triangle Congruence: SSS and SAS 17. Given that HIJK is a rhombus, use SSS to explain why HIL JKL.
HI JK by the definition of a rhombus. HL JL and LI LK because diagonals of a rhombus bisect each other. Therefore, HIL JKL by SSS.
Copyright © by Holt, Rinehart and Winston. All rights reserved.
94
H
I L
K
J
Geometry
CHAPTER 4 REVIEW CONTINUED
PR ; MR QR 18. Given: NR
M
N
Prove: MNR QPR R
Statements
Reasons
1. NR PR; MR QR 1. Given
P
2. MRN QRP
2. Vertical angles are congruent
3. MNR QPR
3. SAS
Q
4-5 Triangle Congruence: ASA, AAS, and HL Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 19. HIK JIK No; HI JI
20. PQR RSP
H K
P
Q
Yes
I
S
J
21. Use ASA to prove the triangles congruent.
R C
A B
bisects ABC and ADC Given: BD Prove: ABD CBD Statements
Reasons
1. BD bisects ABC and ADC
1. Given
2. ABD CBD, ADB CDB
2. Definition of angle bisector
3. BD BD
3. Reflexive property of congruent angles
4. ABD CBD
4. ASA
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95
D
Geometry
CHAPTER 4 REVIEW CONTINUED
4-6 Triangle Congruence: CPCTC
X
YZ , VZ XZ 22. Given: UZ
U
Z
VU Prove: XY
Y
Statements
Reasons
1. UZ YZ, VZ XZ
1. Given
2. UZV YZX
2. Vertical angles are congruent
3. UZV YZX
3. SAS
4. XY VU
4. CPCTC
V
4-7 Introduction to Coordinate Proof Position each figure in the coordinate plane. 23. a right triangle with legs 3 and 4 units in length
24. a rectangle with sides 6 and 8 units in length
y 4
y 8 (0, 6) 6 4 2 (0, 0)
(0, 3)
2 (4, 0) (0, 0)
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Answers will vary. Sample answers shown.
2
x
4
(8, 6)
(8, 0)
x
2 4 6 8
96
Geometry
CHAPTER 4 REVIEW CONTINUED
4-8 Isosceles and Equilateral Triangles 25. Assign coordinates to each vertex and write a coordinate proof. y
Given: rectangle ABCD with diagonals intersecting at z DZ Prove: CZ
1 1 The coordinates of z are 2a, 2c because
C
Z
diagonals of rectangles bisect each other, meaning they intersect at each other’s midpoints. By the midpoint formula the coordinates of z are 0a 0c , . 2 2 1 2 1 1a 2 c 2 CZ (0 12 a)2 (c 2 c) 4 4 1 2 DZ (0 12 a)2 (0 2 c)
1c 2 4
(0, c) B
(0, c)
1
1
( 2 a, 2 c)
D
Ax
(0, 0)
(a, 0)
Therefore,
1a 2 4
CZ DZ, which means CZ DZ. Find each angle measure. A 3x
26. mB 36° C
27. mHEF
E
75°
x
B 110°
H
F
G
28. Given: PQR has coordinates P(0, 0), Q(2a, 0), and R(a, a3 ) Prove: PQR is equilateral.
y R 16
(a, (
3 ) a)
12
PQ (0 2 a)2 (0 0 )2 2a
8
QR (2a a)2 (0 a3 )2 2a
4
RP (a 0 ) ( a3 0) 2a
P
2
2
(2a, 0)
(0, 0) 4
8
12
x
16 Q
Since PQ QR RP, PQR is equilateral.
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97
Geometry
