### 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.

\begin{align}\text{Then}\ \angle XCA &= α\ (\text{alternate angles}\ XY || AB).\\\\ \angle YCB&= β\ (\text{alternate angles}\ XY||AB)\end{align}

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

#### Example 1

Find the value of θ in each of the following.

#### Solution

1. θ + 83° + 63° = 180° (angle sum of ∆ABC)
Hence θ = 34°.
2. θ + 90° + 62° = 180° (angle sum of ∆ABC)
Hence θ = 28°.