Exterior Angles of Triangles

Grade 4

What is an exterior angle?

An exterior angle of a triangle is formed by extending one side of the triangle beyond the vertex. It sits outside the triangle, adjacent to an interior angle.

The key rule: the exterior angle of a triangle equals the sum of the two non-adjacent interior angles. This is sometimes called the exterior angle theorem.

Why does it work?

The three interior angles sum to 180°. The exterior angle and its adjacent interior angle also sum to 180° (angles on a straight line). So the exterior angle must equal what is left over — the other two interior angles.

This is a very quick way to find a missing angle if you know the other two interior angles. No rearranging needed.

Worked example

EXAMPLE

A triangle has interior angles of 52° and 73°. Find the exterior angle at the third vertex.

Exterior angle = 52° + 73° = 125°. Check: interior angle at that vertex = 180° − 125° = 55°; and 52° + 73° + 55° = 180°. ✓

Try this one

The diagram shows two parallel lines and a transversal. Angle p is marked.

Lines \( m \) and \( n \) are parallel. The angle at line \( m \) (below-right of the intersection) is \( 72^\circ \). Find angle \( p \) at line \( n \) (above-left of the intersection). [2]

Nova's hint:

This involves alternate angles formed by a transversal crossing two parallel lines.

Ask yourself: 'Are these angles on opposite sides of the transversal, between the parallel lines?' If two angles form a 'Z' or 'N' shape, what is the relationship between them?

Try to Solve it with Nova?