Math Core

Isosceles triangles show up everywhere, from the gable of a roof to the two sides of a folding ladder. They have a built-in symmetry, and that symmetry gives some of the most useful theorems in geometry. In this lesson you'll prove those theorems with congruent triangles and CPCTC, then use them to find missing angles and sides.

Parts of an isosceles triangle

A triangle is isosceles if it has at least two congruent sides.

The two congruent sides are the legs.
The third side is the base.
The angle formed by the legs is the vertex angle.
The two angles that touch the base are the base angles.

Isosceles triangle ABC with legs AB and AC. The base angles are B and C.

The base angles theorem

Fold an isosceles triangle along the line through its vertex angle and the midpoint of its base, and the two halves land exactly on each other. That's a hint that the base angles match. Here's a proof.

Worked example: Proving the base angles theorem

Given: $\overline{AB} \cong \overline{AC}$

Prove: $\angle B \cong \angle C$

Draw $\overline{AD}$ , the bisector of $\angle BAC$ , meeting $\overline{BC}$ at $D$ . (Every angle has exactly one bisector, so you can always add it.)

AD bisects the vertex angle BAC.

Statement	Reason
1. $\overline{AB} \cong \overline{AC}$	Given
2. $\overline{AD}$ bisects $\angle BAC$	Construction (every angle has a bisector)
3. $\angle BAD \cong \angle CAD$	Definition of angle bisector
4. $\overline{AD} \cong \overline{AD}$	Reflexive property of congruence
5. $\triangle BAD \cong \triangle CAD$	SAS
6. $\angle B \cong \angle C$	CPCTC

Base angles theorem and its converse

Base angles theorem: If two sides of a triangle are congruent, then the angles opposite them are congruent.

Converse: If two angles of a triangle are congruent, then the sides opposite them are congruent.

The converse can be proved with AAS: draw the bisector of the third angle, and the two smaller triangles share that bisector and have two pairs of congruent angles.

The same proof gives a bonus. Since $\triangle BAD \cong \triangle CAD$ , CPCTC also tells you $\overline{BD} \cong \overline{CD}$ and $\angle ADB \cong \angle ADC$ . Those two angles form a linear pair, so each is $90^\circ$ . So the bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

Common mistake

The congruent angles are the ones opposite the congruent sides, not the ones between them. If $\overline{PQ} \cong \overline{PR}$ , the congruent angles are $\angle Q$ and $\angle R$ . Angle $P$ , where the legs meet, is the vertex angle, and it can have any measure. To find the right angles, put your finger on each congruent side and look across the triangle.

Finding angles in isosceles triangles

The base angles are equal and all three angles add to $180^\circ$ . Those two facts solve most problems.

Worked example: From the vertex angle

The vertex angle of an isosceles triangle measures $40^\circ$ . Find each base angle.

The base angles together measure $180^\circ - 40^\circ = 140^\circ$ , and they are equal, so each measures $140^\circ \div 2 = 70^\circ$ .

Worked example: Solving for x

In $\triangle ABC$ , $\overline{AB} \cong \overline{AC}$ , $m\angle B = (3x + 10)^\circ$ and $m\angle C = (5x - 20)^\circ$ . Find $x$ and $m\angle A$ .

$\angle B$ and $\angle C$ are opposite the congruent sides $\overline{AC}$ and $\overline{AB}$ , so they are congruent:

3x + 10 = 5x - 20 \quad\Rightarrow\quad 30 = 2x \quad\Rightarrow\quad x = 15.

Each base angle measures $3(15) + 10 = 55^\circ$ . Then $m\angle A = 180^\circ - 55^\circ - 55^\circ = 70^\circ$ .

Worked example: Using the converse

In $\triangle DEF$ , $m\angle D = m\angle E = 65^\circ$ , $DF = 2y + 1$ and $EF = 4y - 7$ . Find $y$ and $DF$ .

$\angle D$ and $\angle E$ are congruent, so the sides opposite them are congruent. The side opposite $\angle D$ is $\overline{EF}$ , and the side opposite $\angle E$ is $\overline{DF}$ :

2y + 1 = 4y - 7 \quad\Rightarrow\quad 8 = 2y \quad\Rightarrow\quad y = 4.

So $DF = 2(4) + 1 = 9$ . (Check: $EF = 4(4) - 7 = 9$ .)

Equilateral triangles

An equilateral triangle has three congruent sides, and an equiangular triangle has three congruent angles. Apply the base angles theorem to each pair of sides of an equilateral triangle and you get all three angles congruent. The converse works the same way in reverse.

Equilateral and equiangular

A triangle is equilateral if and only if it is equiangular. Each angle of an equilateral triangle measures $180^\circ \div 3 = 60^\circ$ .

An equilateral triangle is also equiangular.

Worked example: An equilateral triangle

$\triangle JKL$ is equilateral with $JK = 3x - 2$ and $KL = x + 6$ . Find the perimeter.

All sides are equal: $3x - 2 = x + 6$ , so $2x = 8$ and $x = 4$ . Each side is $4 + 6 = 10$ , and the perimeter is $3 \cdot 10 = 30$ .

Tip

An equilateral triangle is a special isosceles triangle: any two of its sides can be called the legs. So everything true of isosceles triangles is also true of equilateral ones.

Practice

Practice 1

The vertex angle of an isosceles triangle measures $52^\circ$ . What is the measure of each base angle, in degrees?