Triangle with Two Equal Angles is Isosceles

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.

Proof


Let $$\triangle ABC$$ be a triangle in which $$\angle ABC = \angle ACB$$.

Suppose side $$AB$$ is not equal to side $$AC$$. Then one of them will be greater.

Suppose $$AB > AC$$.

We cut off from $AB$ a length $DB$ equal to $$AC$$.

We draw the line segment $$CD$$.

Since $$DB = AC$$, and $$BC$$ is common, the two sides $$DB, BC$$ are equal to $$AC, CB$$ respectively.

Also, $$\angle DBC = \angle ACB$$.

So by Triangle Side-Angle-Side Equality‎, $$\triangle DBC = \triangle ACB$$.

But $$\triangle DBC$$ is smaller than $$\triangle ACB$$, which is absurd.

Therefore $$AB$$ is not unequal to $$AC$$ and must therefore equal it.

So $$AB = AC$$, which we wanted to prove.

Note
This is Proposition 6 of Book I of Euclid's "The Elements".

Note that it is the converse of Proposition 5: Isosceles Triangles have Two Equal Angles‎.