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{{ printedBook.courseTrack.name }} {{ printedBook.name }} 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.

- $∠ACP≅∠PCB,$ since the angle bisector divides the angle in two equal parts.
- $AC≅BC$ since they are the legs in an isosceles triangle.
- Both contain the side $CP.$

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 |