Proposition I-6

Proposition I-6. 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.
Proof.

Then the triangles ABC and DBC are congruent for the sides AB and CD are equal, the angles ABC and DBC are equal, and they share the common side BC. (Prop 1-4)

This is a contradiction because then the part (triangle DBC) is equal to the whole, which cannot be. (CN 5)

Therefore the sides AB and AC are equal, which was to be proved.

QED

Let triangle ABC be given with angles ABC and ACB equal.

Suppose the sides AB and AC are not equal.

Supposing that AC is the shorter, mark off its length on the side AB.

Label the intersection D. The sides AB and DB are equal.

Then the triangles DBC and ACB are congruent for the sides AC and DB are equal, the angles DBC and ACB are equal, and they share the common side BC. (Prop 1-4)

This is a contradiction because then the part (triangle DBC) is equal to the whole, which cannot be. (CN 5)

Therefore the sides AB and AC are equal, which was to be proved.

QED