Isosceles Triangle

An has two sides of equal length — its two legs. That single piece of symmetry forces a matching symmetry in the angles: the two , sitting at the ends of the third side, always come out equal to each other.

The two legs AB and AC are equal (matching tick marks). Their symmetry forces the two base angles, ∠B and ∠C, to be equal as well.

Explore

Change the apex angle and the leg length and watch the two base angles. As long as the two legs stay equal, what do the base angles always do? Then ask the reverse: if you force the base angles equal, what happens to the sides?

The two legs are always equal — that's the lesson. Reshape with the sliders and watch the base angles ∠B and ∠C stay equal. Then press “Show why ∠B = ∠C”.

Equal side length6.0

Apex angle ∠A40°

The two legs are equal, so the base angles are equal: ∠B = ∠C = 70°, with apex angle ∠A = 40°. The three angles sum to 180°.

An isosceles triangle with apex A and equal legs AB = AC = 6.0. Because the legs are equal, the base angles are equal: ∠B = ∠C = 70° (with apex ∠A = 40°). Press “Show why ∠B = ∠C” to fold the triangle along its axis of symmetry — the line through A and the midpoint of BC — which swaps B and C and lands ∠B exactly onto ∠C.

Make a prediction

Make a prediction: An isosceles triangle has an apex angle (between its two equal sides) of 40°. What does each base angle measure?

70°40°140°100°

Test it

In the tool, drag the apex angle through many values. The two base angles move in lock-step, always equal to each other and always equal to $(180° - \text{apex}) \div 2$ . Now go the other way: the moment the two base angles are equal, the two legs are equal too. This Base Angles Theorem runs both directions.

Why it works

Fold the triangle down its line of symmetry — the line from the apex to the midpoint of the base. The two halves land exactly on top of each other: the two equal legs swap places, and so the two base angles must match. That same mirror explains the converse. If the two base angles are equal, the triangle is symmetric, so the two sides opposite them have to be equal in length. Either piece of equality — sides or base angles — forces the other.

Key result

In an isosceles triangle, the two base angles are equal (the angles opposite the equal sides). It is biconditional: equal sides $\Leftrightarrow$ equal base angles. With apex angle $a$ , each base angle is $\dfrac{180° - a}{2}$ .

Worked example: An isosceles triangle has a base angle of 50°. Find its apex angle, then its other base angle.

The two base angles are equal, so the other base angle is also 50°.
Before the next step: What is the other base angle?
50°80°130°

Isosceles Triangle

Explore

Make a prediction

Test it

Why it works

Practice