Isosceles Triangle Theorem

Theorem: The two angles opposite to the equal sides of an isosceles triangle are equal.

Prerequisites:
Isosceles Triangle (definition)
SAS rule of congruency (proof)

Proof:

Let $\triangle ABC$ be an isosceles triangle with AB = AC. Let AD be the angle bisector of $\angle BAC$.

In triangles $\triangle ABD$ and $\triangle ACD$,
$\qquad\quad\angle BAD = \angle CAD\qquad\qquad\qquad\text{(AD bisects $\angle BAC$)}\\ \qquad\quad AB = AC\qquad\qquad\qquad\qquad\quad\:\!\text{(given)}\\ \qquad\quad AD = AD\qquad\qquad\qquad\qquad\quad\text{(common)}$

Hence by SAS rule of congruency, $\triangle ABD\cong\triangle ACD$.

Therefore, by CPCTC (corresponding parts of the congruent triangles are congruent), $\angle B = \angle C$.

Q.E.D.


Recommended:
Converse of Isosceles Triangle Theorem
Basic Proportionality Theorem
Angle Bisector Theorem