Triangle with Two Equal Angles is Isosceles/Proof 2

Theorem

If a triangle has two angles equal to each other, the sides which subtend the equal angles will also be equal to one another.

Hence, by definition, such a triangle will be isosceles.

In the words of Euclid:

If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another.

(The Elements: Book $\text{I}$: Proposition $6$)

Proof

Let $\angle ABC$ and $\angle ACB$ be the angles that are the same.

\(\text {(1)}: \quad\)

\(\ds \angle ABC\)

\(=\)

\(\ds \angle ACB\)

by hypothesis

\(\text {(2)}: \quad\)

\(\ds BC\)

\(=\)

\(\ds CB\)

Equality is Reflexive

\(\text {(3)}: \quad\)

\(\ds \angle ACB\)

\(=\)

\(\ds \angle ABC\)

by hypothesis

\(\text {(4)}: \quad\)

\(\ds \triangle ABC\)

\(=\)

\(\ds \triangle ACB\)

Triangle Angle-Side-Angle Congruence by $(1)$, $(2)$ and $(3)$

\(\ds AB\)

\(=\)

\(\ds AC\)

from $(4)$

$\blacksquare$