Some Theorems Based on Congruent Triangles

The Isosceles Triangle Theorems

The Isosceles TriangleA triangle is isosceles iff it has at least two sides congruent.

vertex angle

legleg

base angles

base

Parts of an isosceles triangle:

Theorem 4-1 The Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.– Can also be stated: The base angles of an

isosceles triangle are congruent.

Corollaries to the Isosceles Triangle Theorem

Corollary 1– An equilateral triangle is also equiangular.

Corollary 2– An equilateral triangle has three 60° angles.

Corollary 3– The bisector of the vertex angle of an isosceles

triangle is perpendicular to the base at its midpoint.

Theorem 4-2

If the base angles of a triangle are congruent, then the sides opposite those angles are congruent.

Corollary to the Converse of the Isosceles Triangle Theorem

Corollary 1– An equiangular triangle is also equilateral.