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3.7 Angle-Side Theorems
Objectives: • Apply theorems relating the angle measures
and the side lengths of triangles
Theorem: If two sides of a triangle are congruent, then the opposite angles are congruent.
Theorem: If two angles of a triangle are congruent, then the opposite sides are congruent.
A
B
CR
S
T
Theorem: If two sides of a triangle are congruent, then the opposite angles are congruent.
Theorem: If two angles of a triangle are congruent, then the opposite sides are congruent.
How do you prove triangles are isosceles?
1. Prove at least two sides congruent.
2. Prove at least two angles congruent.
Based on these last theorems, what conclusions can be drawn about equilateral and equiangular triangles?
A
B
C X
Y
Z
Theorem: If two sides of a triangle are not congruent, then the opposite angles are not congruent and the larger angle is opposite the longer side.
Theorem: If two angles of a triangle are not congruent, then the opposite sides are not congruent and the longer side is opposite the larger angle.
A
C
B
Z Y
X
FE
D
Example 1: List the sides from least to greatest.
24° 46°
110°
NL
M
13
Example 2: List the angles from least to greatest.
12
5
Example 3:
Statements Reasons
1. 1.
2. 2.
3. 3. Definition of isoscelesKMN N
Given
All radii of a circle are congruent
K
KN KM
Given :
Pr ove :
K
KMN N
L
M N
K
Example 4:
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
8. 8.
9. 9.
Linear Pair Postulate2 is supp. to IGH
Given
Linear Pair Postulate
1 2
1 is supp. to KLJ
Given : 1 2, ,
Prove : is an isosceles Δ
LJ GI KL HG
FKH
F
I
G
HJ
L
K
21
Congruent Supplements Thm.KLJ IGH
GivenLJ JI
SASKLJ HGI
GivenKL HG
CPCTCK H
Definition of isosceles is isoscelesKFH
Example 4:
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
Definition of isoscelesA E
Given
Definition of isosceles
is an isos. with baseFAE AE
AF FE
GivenAB DE
Addition PropertyAD BE
Reflexive PropertyBD BD
SASFAD FED
Given: is an isosceles Δ with base
Prove:
FAE AE
AB DE
FAD FEB
F
A EB D
