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Isosceles Triangle Theorem

Proof

Base Angles Theorem

If two sides in a triangle are congruent, then the angles opposite them are congruent.

This theorem will be proven using congruent triangles.
has two congruent sides.

Draw the angle bisector that bisects and intersects at point

and share the following features.

  • since the angle bisector divides the angle in two equal parts.
  • since they are the legs in an isosceles triangle.
  • Both contain the side

For both triangles, two sides and the included angle are congruent. Thus, according to the Side-Angle-Side Congruence Theorem. Because and are corresponding angles in congruent triangles, they are congruent. Therefore, if two sides in a triangle are congruent, the angles opposite them are congruent.
This can be summarized in a two-column proof.

Statement Reason
Given
Draw an angle bisector, Construction of angle bisector
Definition of angle bisector
Common side
Side-Angle-Side Congruence Theorem
Corresponding parts of congruent triangles are congruent