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°
Write a two-column proof.
Given:
Prove:
Proof:
ReasonsStatements
3. Def. of isosceles 3. ABC and BCD are isosceles
1. Given1.
6. 6. Substitution
5. 5. Given
4. 4. Isosceles Theorem
2. Def. of segments2.
Write a two-column proof.
Given: .
Prove:
Proof:
ReasonsStatements
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
Multiple-Choice Test Item If and what is the measure of
Read the Test Item CDE is isosceles with base Likewise, CBA is isosceles with
A. 45.5 B. 57.5 C. 68.5 D. 75
Solve the Test Item
Angle Sum Theorem
Substitution
Add.
Subtract 120 from each side.
Divide each side by 2.
Step 1 The base angles of CDE are congruent. Let
Def. of vertical angles
Substitution
Add.
Step 2 are vertical angles so they have equal measures.
Step 3 The base angles of CBA are congruent.
Angle Sum Theorem
Substitution
Subtract 30 from each side.
Divide each side by 2.
Answer: D
Answer: A
Multiple-Choice Test Item If and what is the measure of
A. 25 B. 35 C. 50 D. 130
Answer:
Name two congruent angles.
Answer:
Name two congruent segments.
By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So,
a. Name two congruent angles.
Answer:
Answer:
b. Name two congruent segments.
Since the angle was bisected,
Each angle of an equilateral triangle measures 60°.
EFG is equilateral, and bisects bisectsFind and
Answer:
Add.
Exterior Angle Theorem
Substitution
is an exterior angle of EGJ.
Subtract 75 from each side.
Linear pairs are supplementary.
Substitution
Answer: 105
EFG is equilateral, and bisects bisectsFind
a. Find x.
b.
Answer: 90
Answer: 30
ABC is an equilateral triangle. bisects