1. Lesson 4-6
Isosceles Triangles

2. Standardized Test Practice:
Transparency 4-6
5-Minute Check on Lesson 4-5
Refer to the figure. Complete each congruence statement and the postulate or theorem that applies.
1. WXY  _____ by _____.
2. WYZ  _____ by _____.
3. VWZ  _____ by _____.
What additional congruence statement is necessary to prove RST  UVW by the ASA Postulate?
Standardized Test Practice: A
T  W B
R  U C
ST  UW D
RT  VW

3. Standardized Test Practice:
Transparency 4-6
5-Minute Check on Lesson 4-5
Refer to the figure. Complete each congruence statement and the postulate or theorem that applies.
1. WXY   VZY by ASA . or AAS
2. WYZ   VYX by AAS .
3. VWZ   WVX by ASA . AAS, SSS
or SAS
What additional congruence statement is necessary to prove RST  UVW by the ASA Postulate?
Standardized Test Practice: A
T  W B
R  U C
ST  UW D
RT  VW

4. Objectives Use properties of isosceles triangles
Use properties of equilateral triangles

5. Vocabulary Vertex angle – the angle formed by the two congruent sides
Base angle – the angle formed by the base and one of the congruent sides

6. Theorems
Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Converse of Isosceles Triangle Theorem: If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

7. Corollaries
A triangle is equilateral if, and only if, it is equiangular
Each angle of an equilateral triangle measures 60°

8. Isosceles Triangle A + B + C = 180° B Vertex angle leg leg A base C
Base Angles A  C
A + B + C = 180°

9. Write a two-column proof.
Given:
Prove:
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

10. Write a two-column proof.
Given:
Prove:
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

11. Multiple-Choice Test Item
If and what is the measure of
A B. 57.5
C D. 75
Read the Test Item
CDE is isosceles with base Likewise, CBA is isosceles with
Solve the Test Item
Step 1 The base angles of CDE are congruent. Let

12. Angle Sum Theorem
Substitution
Add.
Subtract 120 from each side.
Divide each side by 2.
Step are vertical angles so they have equal measures.
Def. of vertical angles
Substitution

13. Step 3 The base angles of CBA are congruent.
Angle Sum Theorem
Substitution
Add.
Subtract 30 from each side.
Divide each side by 2.
Answer: D

14. Multiple-Choice Test Item
If and what is the measure of
A B C D. 130
Answer: A

15. Name two congruent angles.
Answer:

16. Name two congruent segments.
By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So,
Answer:

17. a. Name two congruent angles.
Answer:
b. Name two congruent segments.
Answer:

18. ABC is an equilateral triangle. bisects
a. Find x.
Answer: 30 b.
Answer: 90

19. Summary & Homework Summary: Homework:
Two sides of a triangle are congruent if, and only if, the angles opposite those sides are congruent.
A triangle is equilateral if, and only if, it is equiangular.
Homework:
pg : 9, 10, 13-18, 27