If two angles in a triangle are equal, then the triangle is isosceles

From ProofWiki

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:

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

Proof 1

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 Equality‎, $\triangle DBC = \triangle ACB$.

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

Therefore, have $AB \le AC$.

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

$\blacksquare$

Proof 2

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

$\blacksquare$

Sources

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An isosceles triangle is a triangle with two sides that are equal in length. This means that two angle will also be equal to each other. Is there any way that a triangle could have two equal angles, but not be an isosceles triangle?

If two sides of a triangle are congruent , then the angles opposite to these sides are congruent.

∠ P ≅ ∠ Q

Proof:

Let S be the midpoint of P Q ¯ .

Join R and S .

Since S is the midpoint of  P Q ¯ , P S ¯ ≅ Q S ¯ .

By Reflexive Property ,

R S ¯ ≅ R S ¯

It is given that P R ¯ ≅ R Q ¯

Therefore, by SSS ,

Δ P R S ≅ Δ Q R S

Since corresponding parts of congruent triangles are congruent,

∠ P ≅ ∠ Q

The converse of the Isosceles Triangle Theorem is also true.

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

If ∠ A ≅ ∠ B , then A C ¯ ≅ B C ¯ .