Triangle with Two Equal Angles is Isosceles/Proof 1

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 $\triangle ABC$ 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.

Without loss of generality, 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 Congruence:

$\triangle DBC = \triangle ACB$

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

Therefore, have $AB \le AC$.

A similar argument shows the converse, and hence $AB = AC$.

$\blacksquare$

Historical Note

This proof is Proposition $6$ of Book $\text{I}$ of Euclid's The Elements.
It is the converse of Proposition $5$: Isosceles Triangle has Two Equal Angles‎.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 1 (2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions