⚹   An Angle Outside the Triangle

Extend one side of a triangle past a vertex. The angle formed between the extended line and the next side is called the exterior angle.

We'll work through four triangles. The first two guide you step by step; the last two ask you to predict. By the end, you'll have worked out the rule yourself.

⚹   Using the Rule

Now apply what you know. For each triangle, use the exterior-angle rule to find the missing value.

Sometimes you'll work forwards (find exterior from two interiors). Sometimes you'll work backwards (find a missing interior, given the exterior and one other interior).

⚹   Combine with Other Facts

A-grade work: combining the exterior-angle rule with other triangle facts you know.

The triangles below are isosceles, equilateral, or right-angled. You'll need to use that information plus the exterior-angle rule to find the missing angle.

Remember: isosceles triangles have two equal base angles. Equilateral triangles have three 60° angles. Right-angled triangles have one 90° angle.

⚹   Four-Sided Figures

You already know a triangle's angles sum to 180°. What about a quadrilateral?

We'll work through four shapes. The first two guide you step by step; the last two ask you to predict. By the end you'll have worked out the rule yourself.

Diagrams are not to scale — trust the numbers.

⚹   The Pattern of Triangles

A quadrilateral splits into 2 triangles — hence 2 × 180° = 360°.

What about a pentagon? A hexagon? A heptagon? Pick any vertex and draw all the diagonals to the non-adjacent vertices. Count the triangles.

Fill in the table. Each entry follows from the one above.

⚹   Apply the Rule

Now use (n − 2) × 180° to solve problems. Some questions ask for the angle sum. Others give a polygon with one missing angle — use the rule to find it.

⚹   Equal Sides, Equal Angles

A regular polygon has all sides equal AND all angles equal.

If you know the angle sum of an n-gon is (n − 2) × 180°, and all n angles are equal, then each interior angle is:

(n − 2) × 180° ÷ n

Find the interior angle for each regular polygon below.

⚹   A Surprising Fact

Here's something you might not expect. Look at the regular pentagon below. Each exterior angle is 72°.

Add them all up: 5 × 72° = ?

For the hexagon, each exterior is 60°. 6 × 60° = ?

For the octagon, each exterior is 45°. 8 × 45° = ?

Work through these three and watch for the pattern.

⚹   Each Exterior of a Regular Polygon

Since the n exterior angles of a regular polygon are all equal and sum to 360°, each one is:

360° ÷ n

This is a faster way to find the interior angle too: interior + exterior = 180° (they sit on a straight line at each vertex).

⚹   Working Backwards

A-grade work: you're told the interior angle (or exterior angle) of a regular polygon. You find how many sides it has.

Strategy: use 360° ÷ exterior = n. If given the interior, first subtract from 180° to get the exterior.

Example: a regular polygon has interior angle 108°. What is it? Exterior = 180° − 108° = 72°. n = 360° ÷ 72° = 5. It's a pentagon.