How to find the base of an isosceles triangle if the sides and the angle between them are known?

1. Suppose ABC is an isosceles triangle with lateral sides AB and BC. Let them be 10.

Let the angle B between them be equal to 120 °.

2. The sides and angles in any triangle are related by the ratio: (in this case) AB / sin C = BC / sin A = AC / sin A (theorem of sines).

3. Find the angles at the base of an isosceles triangle: (180 – 120) / 2 = 30.

4. Substitute the data into the equation: AB / sin 120 ° = 10 / sin 30 ° = 10 / sin 30 °.

AB = (10 × sin 120 °) / sin 30 ° = (10 × √3 / 2) / 0.5 = 5√3 × 0.5 = 2.5√3.