### Proving that the sum of the interior angles of a triangle is 180°

The statement of the result is called a **theorem**. This comes from a Greek word meaning 'a thing to be gazed upon' or 'a thing contemplated by the mind'. The word 'theatre' comes from the same Greek root.

#### Theorem

The sum of the angles of a triangle is 180°.

#### Proof

Let ∆*ABC *be a triangle.

Let \(\angle\)*BAC* = α, \(\angle\)*CBA* = β and \(\angle\)*BCA *= γ.

We must prove that α + β + γ = 180°.

The line *XY* is drawn parallel to the line *AB* through the vertex *C*.

Hence α + β + γ = 180° (straight angle at *C*).

#### Example 1

Find the value of θ in each of the following.

#### Solution

- θ + 83° + 63° = 180° (angle sum of ∆
*ABC*)

Hence θ = 34°. - θ + 90° + 62° = 180° (angle sum of ∆
*ABC*)

Hence θ = 28°.