**Introduction of triangle with two equal angles:**

The triangles with two equal angles are called isosceles triangles. In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the equal measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem. Some mathematicians define isosceles triangles to have simply two equal sides, whereas others define that an isosceles triangle is one with at least two equal sides. The latter definition would make every equilateral triangles isosceles triangles. (Source: Wikipedia)

Clearly all equilateral triangles also have all the properties of an isosceles triangle.

**Properties:**

- The unequal side of an isosceles triangle is usually known to as the 'base' of the triangle.

- The base angles of an isosceles triangle are always equal. In the figure above, the angles ∠ABC and ∠ACB are always the same.

- When the 3rd angle is a right angle, it is called a "right isosceles triangle".

- The altitude is a perpendicular bisector from the base to the uppermost vertex.

**Solving an isosceles triangle:**

The base, leg or altitude of an isosceles triangle is found if you know the other two. Perpendicular bisector of the base makes an altitude of the triangle. These form two congruent right triangles that can be solved using Pythagoras' Theorem as shown below.

**Base:**

To find the base given the leg and altitude, use the formula:

base = 2√L^{2} - A^{2}

where, L is the length of a leg

A is the altitude

**Leg:**

To find the leg length given the base and altitude, use the formula:

Leg = √A^{2} + (B/2)^{2}

where, B is the length of the base

A is the altitude

**Altitude:**

To find the altitude of an isosceles triangle given the base and leg, use the formula:

Altitute = √L^{2} - (B/2)^{2}

where, B is the length of the base

L is the length of a leg

**Example 1:**

Find the altitude of the triangle base = 6cm and leg = 7cm.

**Solution:**

Given: Base = 6cm leg = 7cm

Altitute = √L^{2} - (B/2)^{2}

= √7^{2} - (6/2)^{2}

= √49 - 3^{2}

= √49 - 9

= √40

= 6.32cm

**Example 2:**

Find the base of a triangle, leg = 12cm and altitude = 10cm

**Solution:**

Given: A = 10cm L = 12cm

base = 2√L^{2} - A^{2}

= 2√12^{2} - 10^{2}

= 2√144 -100

= 2√44

= 13.26 cm.

These are some examples triangle with two equal angles.