If two sides in a triangle are congruent, then the angles opposite them are congruent.
This theorem will be proven using congruent triangles.
△ABC has two congruent sides.
Draw the angle bisector that bisects ∠C and intersects AB at point P.
△ACP and △BCP share the following features.
For both triangles, two sides and the included angle are congruent. Thus, △ACP≅△BCP
according to the Side-Angle-Side Congruence Theorem. Because ∠PAC and ∠PBC 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 |
AC≅CB | Given |
Draw an angle bisector, CP. | Construction of angle bisector |
∠ACP≅∠PCB | Definition of angle bisector |
CP≅CP | Common side |
ΔACP≅ΔBCP | Side-Angle-Side Congruence Theorem |
∠B≅∠C | Corresponding parts of congruent triangles are congruent |