# Isosceles triangle

An isosceles triangle is a triangle that has at least two sides of equal length. Two examples are given in the figure below.

## Parts of an isosceles triangle

For an isosceles triangle with only two congruent sides, the congruent sides are called legs. The third side is called the base. The angle opposite the base is called the vertex angle, and the angles opposite the legs are called base angles.

## Lengths of an isosceles triangle

The height of an isosceles triangle is the perpendicular line segment drawn from base of the triangle to the opposing vertex.

Using the Pythagorean Theorem, we can find that the base, legs, and height of an isosceles triangle have the following relationships:

## Base angles of an isosceles triangle

The base angles of an isosceles triangle are the same in measure. Refer to triangle ABC below.

AB ≅AC so triangle ABC is isosceles. ABC can be divided into two congruent triangles by drawing line segment AD, which is also the height of triangle ABC.

Using the Pythagorean Theorem where l is the length of the legs, . Based on this, △ADB≅△ADC by the Side-Side-Side theorem for congruent triangles since BD ≅CD, AB ≅ AC, and AD ≅AD. So, ∠B≅∠C, since corresponding parts of congruent triangles are also congruent.

## Symmetry in an isosceles triangle

The altitude of an isosceles triangle is also a line of symmetry.

Leg AB reflects across altitude AD to leg AC. Similarly, leg AC reflects to leg AB. Base BC reflects onto itself when reflecting across the altitude.

## 45-45-90 triangles

When the base angles of an isosceles triangle are 45°, the triangle is a special triangle called a 45°-45°-90° triangle. The length of the base, called the hypotenuse of the triangle, is times the length of its leg.

## Apothem of a regular polygon

The apothem of a regular polygon is also the height of an isosceles triangle formed by the center and a side of the polygon, as shown in the figure below.

For the regular pentagon ABCDE above, the height of isosceles triangle BCG is an apothem of the polygon.