4.6 isosceles triangles

4.6 Isosceles Triangles

Objectives

Use properties of isosceles triangles

Use properties of equilateral triangles

Properties of Isosceles Triangles

The formed by the ≅ sides is called the vertex angle.

The two ≅ sides are called legs. The third side is called the base.

The two s formed by the base and the legs are called thebase angles.

leg leg

base

vertex

Isosceles Triangle Theorem

Theorem 4.9If two sides of a ∆ are ≅, then the s opposite those sides are ≅ (if AC ≅ AB, then B ≅ C). A

B C

Write a two-column proof.

Given:

Prove:

Example 1:

Proof:

ReasonsStatements

3. Def. of Isosceles 3. ABC and BCD are isosceles triangles

1. Given1.

6. 6. Substitution

5. 5. Given

4. 4. Isosceles Theorem

2. Def. of Segments2.

Example 1:

Write a two-column proof.

Given: .

Prove:

Your Turn:

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

Your Turn:

The Converse of Isosceles Triangle Theorem

Theorem 4.10

If two s of a ∆ are ≅, then the sides opposite those s are ≅ (if B ≅ C, then AC ≅ AB).

Answer:

Name two congruent angles.

Example 2:

Answer:

Name two congruent segments.

By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So,

Example 2:

a. Name two congruent angles.

Answer:

Answer:

b. Name two congruent segments.

Your Turn:

Properties of Equilateral ∆s

Corollary 4.3A ∆ is equilateral iff it is equiangular.

Corollary 4.4Each of an equilateral ∆ measures 60°.

Since the angle was bisected,Each angle of an equilateral triangle measures 60°.

EFG is equilateral, and bisects bisectsFind and

Example 3a:

Answer:

Add.

Exterior Angle Theorem

Substitution

is an exterior angle of EGJ.

Example 3a:

Subtract 75 from each side.

Linear pairs are supplementary.

Substitution

Answer: 105

EFG is equilateral, and bisects bisectsFind

Example 3b:

a. Find x.

b.

Answer: 90

Answer: 30

ABC is an equilateral triangle. bisects

Your Turn:

Assignment

Geometry:Pg. 219 #9 – 28, 35 - 37

Pre-AP Geometry: Pg. 219 #9 – 30, 35 – 37, & 40