1. 4.6 Isosceles Triangles Theorem 4.9 Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Theorem 4.10
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Corollaries
4.3 A triangle is equilateral if and only if it is equiangular
4.4 Each angle of an equilateral triangle measures 60°

2. Write a two-column proof.
Given:
Prove:
Example 6-1a

3. 3. ABC and BCD are isosceles
Proof:
Reasons
Statements
1. Given 1.
2. Def. of segments 2.
3. Def. of isosceles 
3. ABC and BCD are isosceles 5.
5. Given 4.
4. Isosceles  Theorem 6.
6. Substitution
Example 6-1b

4. Write a two-column proof.
Given:
Prove:
Example 6-1c

5. 2. Def. of isosceles triangles 1.
Proof:
Reasons
Statements
1. Given
3. Isosceles  Theorem
2. Def. of isosceles triangles 1.
2. ADB is isosceles. 3. 4. 5.
4. Given
5. Def. of midpoint
6. SAS 7.
7. CPCTC
6. ABC ADC
Example 6-1d

6. Multiple-Choice Test Item If and what is the measure of
A B C D. 75
Read the Test Item
CDE is isosceles with base Likewise, CBA is isosceles with
Example 6-2a

7. Step 1 The base angles of CDE are congruent. Let
Solve the Test Item
Step 1 The base angles of CDE are congruent. Let
Angle Sum Theorem
Substitution
Add.
Subtract 120 from each side.
Divide each side by 2.
Example 6-2b

8. Step 2 are vertical angles so they have equal measures.
Def. of vertical angles
Substitution
Step 3 The base angles of CBA are congruent.
Angle Sum Theorem
Substitution
Add.
Subtract 30 from each side.
Divide each side by 2.
Example 6-2c

9. Answer: D
Example 6-2d

10. Multiple-Choice Test Item If and what is the measure of
A B C D. 130
Answer: A
Example 6-2e

11. Name two congruent angles.
Answer:
Example 6-3a

12. Name two congruent segments.
By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So,
Answer:
Example 6-3b

13. a. Name two congruent angles.
Answer:
b. Name two congruent segments.
Answer:
Example 6-3c

14. EFG is equilateral, and bisects bisects Find and
Since the angle was bisected,
Each angle of an equilateral triangle measures 60°.
Example 6-4a

15. is an exterior angle of EGJ.
Exterior Angle Theorem
Substitution
Add.
Answer:
Example 6-4b

16. EFG is equilateral, and bisects bisects Find
Linear pairs are supplementary.
Substitution
Subtract 75 from each side.
Answer: 105
Example 6-4c

17. ABC is an equilateral triangle. bisects
a. Find x.
Answer: 30 b.
Answer: 90
Example 6-4d