Equal sides give equal base angles

Two sides of a triangle are equal. Must the base angles, opposite those sides, match?

The rule

Yes, the angles opposite the equal sides match.

An isosceles triangle has two equal sides, AB and AC. They meet at the apex A. The third side, BC, is the base. The angles at B and C are the base angles.

Draw the bisector of angle A, the line that halves it. It meets the base BC at D. Compare triangles ABD and ACD.

• AB = AC, given.
• Angle BAD = angle CAD, by the bisector.
• AD is shared.

That is SAS: two sides and the angle between them. The two triangles match. Matching triangles have equal matching angles. So angle B = angle C.

Change the apex. Switch off Both sides equal. Compare the base angles.