4-5 Isosceles and Equilateral Triangles

Isosceles Triangles

• The congruent sides of an isosceles triangle are its legs.

• The third side is the base.• The legs form the vertex angle.• The other two angles are the base angles.

Isosceles Triangle Theorems

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

(Also known as the Base Angles Theorem)

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

Using the Isosceles Triangle Theorems

• Is AB congruent to CB? Explain.

• Is A congruent to DEA? Explain.

Answer the following:

• Is WVS congruent to S? Explain.

• Is TR congruent to TS? Explain.

Bisectors

Theorem 4-5: If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base.

Using Algebra

• What is the value of x?

Suppose mA = 27. What is the value of x?

Corollaries

• A corollary is a theorem that can be proved easily using another theorem.

Corollary to Theorem 4-3: If a triangle is equilateral, then the triangle is equiangular.

Corollary to Theorem 4-4: If a triangle is equiangular, then the triangle is equilateral.

Finding Angle Measures

• What are the measures of A, B, and ADC?