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