Isosceles & Equilateral Triangle Theorems

An isosceles triangle has at least two congruent sides, called the legs. The third side is the base, the angle between the legs is the vertex angle, and the two angles that touch the base are the base angles. The whole topic runs on one matched pair of facts about those parts.

The Base Angles Theorem: if two sides of a triangle are congruent, then the angles opposite those sides are congruent. Its converse runs the other way: if two angles are congruent, the sides opposite them are congruent. Combine either one with the fact that a triangle's angles total $180^\circ$ and most problems become a one-variable equation.

The Base Angles Theorem and its converse

Congruent sides face congruent angles. In $\triangle ABC$ with $\overline{AB} \cong \overline{AC}$ , the congruent angles are the ones opposite those sides: $\angle C$ (opposite $\overline{AB}$ ) and $\angle B$ (opposite $\overline{AC}$ ). Those are the base angles.

The converse lets you travel in reverse. If a triangle has two congruent angles, then the sides opposite those angles are congruent — so the triangle is isosceles even if no side lengths were given.

Be careful locating the base angles when the triangle is rotated. The base is not "the bottom" — it is the side that is not one of the two congruent legs, and the base angles are the two angles the legs do not enclose. In the figure below, the two congruent legs carry matching tick marks and the base angles carry matching arcs.

Equilateral means equiangular

Apply the Base Angles Theorem to all three pairs of sides of an equilateral triangle and every angle comes out congruent to every other. Three congruent angles summing to $180^\circ$ forces each one to measure $60^\circ$ .

The converse holds too: a triangle with three congruent angles has three congruent sides. Equilateral and equiangular are two views of the same triangle.

Setting up the equation

Nearly every numeric problem is the same equation in costume. Call the vertex angle $v$ and each base angle $b$ . The Triangle Angle-Sum Theorem gives $v + 2b = 180$ .

Given the vertex angle, subtract it from $180$ and divide by $2$ to get each base angle. Given a base angle, double it and subtract from $180$ to get the vertex angle.

Worked examples

Example 1: vertex angle given

The vertex angle of an isosceles triangle measures $36^\circ$ . Find each base angle.

Let each base angle measure

b

and use the Triangle Angle-Sum Theorem

36 + 2b = 180

Subtract

36

from both sides

2b = 144

Divide both sides by

2

b = 72

Answer: Each base angle measures $72^\circ$ .

Example 2: base angle given

In $\triangle ABC$ , $\overline{AB} \cong \overline{AC}$ and $m\angle B = 65^\circ$ . Find $m\angle A$ .

The base angles are opposite the congruent sides, so

\angle C \cong \angle B

m\angle C = 65^\circ

Use the Triangle Angle-Sum Theorem for the vertex angle

m\angle A = 180^\circ - 65^\circ - 65^\circ

Simplify

m\angle A = 50^\circ

Answer: $m\angle A = 50^\circ$

Example 3: solving for a variable

An isosceles triangle has a vertex angle of $30^\circ$ , and each base angle measures $2x + 5^\circ$ . Find $x$ .

Vertex angle plus two base angles equals

180^\circ

30 + 2(2x + 5) = 180

Subtract

30

from both sides

2(2x + 5) = 150

Divide both sides by

2

2x + 5 = 75

Subtract

5

, then divide by

2

x = 35

Answer: $x = 35$

Try one yourself

Common questions

Which angles are the base angles?

The two angles opposite the congruent sides. Find the legs first, then look across from each leg — those two angles are congruent. Do not assume the base angles are at the bottom of the picture; the triangle may be rotated.

Can a triangle be both isosceles and right?

Yes. A right isosceles triangle has a $90^\circ$ vertex angle and two $45^\circ$ base angles, since $90 + 2(45) = 180$ . Its two legs are the congruent sides.

Is an equilateral triangle also isosceles?

Yes. Isosceles requires at least two congruent sides, and an equilateral triangle has three. Every equilateral triangle is isosceles, but not the other way around.

How do I prove a triangle is isosceles if I only know angles?

Use the converse of the Base Angles Theorem: show two angles are congruent, and the sides opposite them must be congruent. Two matching angle measures are enough.

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Isosceles & Equilateral Triangle Theorems

The Base Angles Theorem and its converse

Equilateral means equiangular

Setting up the equation

Worked examples

Example 1: vertex angle given

Example 2: base angle given

Example 3: solving for a variable

Try one yourself

Common questions

Keep learning

Angles of Triangles

Triangle Congruence Proofs

Congruent Figures & Corresponding Parts (CPCTC)

Classifying Triangles & Quadrilaterals by Coordinates