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Isosceles triangles and equal base angles

2026-08-28Plane GeometryMs Ho

Equal sides give equal base angles; use that to finish missing-angle problems quickly.

Key formulas

AB = AC ⇒ ∠ABC = ∠ACB
sum of angles in a triangle = 180°

Key property

In an isosceles triangle, the angles opposite the equal sides are equal. Mark the equal sides first so the equal angles are obvious.

Worked example

In △ABC, AB = AC and ∠BAC = 40°. The base angles are equal, so each is (180° − 40°) / 2 = 70°. Therefore ∠ABC = ∠ACB = 70°.

Exterior angle follow-up

If side BC is extended to D, the exterior angle ACD equals ∠ABC = 70° by the exterior angle theorem (equal to the remote interior angle when the diagram fits), or compute 180° − 70° = 110° if ACD is adjacent on a straight line. Read which angle is asked.

Worked example

Worked example

Problem. In isosceles △ABC with AB = AC, the vertex angle ∠A = 40°. Find the base angles.

Solution. Base angles are equal: ∠B = ∠C. Angle sum: 40° + 2∠B = 180° → 2∠B = 140° → ∠B = ∠C = 70°.

Vertex angle unknown

Base angles each 55° → vertex angle = 180° − 110° = 70°.

Review quiz

Isosceles with vertex angle 50°. Each base angle is:

Practice questions

  1. AB = AC and ∠ABC = 65°. Find ∠BAC.
  2. An isosceles triangle has vertex angle 20°. Find each base angle.

Study notes

Which angles are the base angles?

The two angles adjacent to the base, opposite the equal sides.

Can an isosceles triangle be right-angled?

Yes. Base angles 45° and 45° with vertex 90° is the familiar isosceles right triangle.

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