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LESSON
Chapter Review
4
4-1 Classifying Triangles
W
Classify each triangle by its angle measure.
Z
1. nXYZ
2. nXYW
3. nXZW
25
35
X
Classify each triangle by its side lengths.
4. nDEF
60
2
5. nDEG
Y
D
G
6. nEFG
8
10
E
F
4-2 Angle Relationships in Triangles
Find each angle measure.
D
7. m ACB
4x – 45
A
8. m K
K
C
2x – 3
x–7
M
B
25° 3x 2x
N
L
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.
4-3 Congruent Triangles
Given nABC ù nXYZ. Identify the congruent corresponding parts.
10. BC ù
11. ZX ù
12.
13.
Aù
L
Given nJKL ù nPQR. Find each value.
14. x
Yù
42°
15. RP
J
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R
3y – 2
K
2y – 1
22
3x°
Q
P
Geometry
CHAPTER 4 REVIEW CONTINUED
16. Given: < zz k; BD ù CD ; AB ù AC ; AD ⊥ CB ; AD ⊥ XW ;
XAC ù WAB
x
Prove: nABD ù nACD
k
y
z
C
Statements
D
1.
2. AD ù AD
2.
3. < zz k; AD ⊥ CB ; AD ⊥ XW
3.
4.
4. Def. of ⊥ lines
ADB ù
5.
ADC
6. Given
6.
7.
XAC ù ACD;
WAB ù ABD
7.
8. Transitive Property of
Congruence
8.
9.
B
Reasons
1. BD ù CD ; AB ù AC ;
5.
w
A
CAD ù
9.
BAD
10.
10. Def of Congruent Triangles
4-4 Triangle Congruence: SSS and SAS
17. Given that HIJK is a rhombus, use SSS to explain
why nHIL ù nJKL.
H
L
K
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I
J
Geometry
CHAPTER 4 REVIEW CONTINUED
18. Given: NR ù PR ; MR ù QR
M
N
Prove: nMNR ù nQPR
R
P
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. nHIK ù nJIK
20. nPQR ù nRSP
H
K
P
Q
I
S
J
21. Use ASA to prove the triangles congruent.
R
C
A
B
Given: BD bisects
ABC and
ADC
Prove: nABD ù nCBD
D
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Geometry
CHAPTER 4 REVIEW CONTINUED
4-6 Triangle Congruence: CPCTC
X
U
Z
22. Given: UZ ù YZ , VZ ù XZ
Y
Prove: XY ù VU
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
y
4
8
6
4
2
2
x
2
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4
x
2 4 6 8
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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
Prove: CZ ù DZ
x
Find each angle measure.
A
3x
26. m B
C
x
27. m HEF
E
B
110°
H
F
G
28. Given: nPQR has coordinates P(0, 0), Q(2a, 0), and R(a, aÏ3
w)
Prove: nPQR is equilateral.
y
16
12
8
4
x
P
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8
12
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Geometry
CHAPTER
Postulates and Theorems
4
Theorem 4-2-1
Corollary 4-2-2
Corollary 4-2-3
(Triangle Sum Theorem) The sum of the angle measures of
a triangle is 180°. m A m B m C ! 180°
The acute angles of a right triangle are complementary.
The measure of each angle of an equilateral triangle is 60°.
Theorem 4-2-4
(Exterior Angle Theorem) The measure on an exterior angle
of a triangle is equal to the sum of the measures of its
remote interior angles.
Theorem 4-2-5
(Third Angles Theorem) If two angles of one triangle are
congruent to two angles of another triangle, then the third
pair of angles are congruent.
(Side-Side-Side (SSS) Congruence) If three sides of one triangle are congruent to three sides of another triangle, then
the triangles are congruent.
Postulate 4-4-1
Postulate 4-4-2
Postulate 4-5-1
Theorem 4-5-2
Theorem 4-5-3
Theorem 4-8-1
Converse 4-8-2
Corollary 4-8-3
Corollary 4-8-4
(Side-Angle-Side (SAS) Congruence) If two sides and the
included angle of one triangle are congruent to two sides
and the included angle of another triangle, then the triangles are congruent.
(Angle-Side-Angle (ASA) Congruence) 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.
(Angle-Angle-Side (AAS) Congruence) If two angles and the
nonincluded side of one triangle are congruent to two
angles and the nonincluded side of another triangle, then
the triangles are congruent.
(Hypotenuse-Leg (HL) Congruence) If the hypotenuse and
a leg of a right triangle are congruent to the hypotenuse
and a leg of another right triangle, then the triangles are
congruent.
(Isosceles Triangle Theorem) If two sides of a triangle
are congruent, then the angles opposite the sides are
congruent.
(Converse of Isosceles Triangle Theorem) If two sides of a
triangle are congruent, then the angles opposite the sides
are congruent.
(Equilateral Triangle) If a triangle is equilateral, then it is
equiangular.
(Equiangular Triangle) If a triangle is equiangular, then it is
equilateral.
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Geometry