Angles, parallel lines and similar triangles

Use angle facts, parallel-line rules, polygon angle sums and similar triangles to find missing angles and lengths.

Also known as: angles in a triangle, alternate angles, corresponding angles, similar triangles, polygons.

SAT · Geometry and Trigonometry GCSE Maths · Geometry and measures IGCSE Maths · Geometry

Key ideas

Angles on a straight line sum to 180°; angles around a point sum to 360°.
Angles in a triangle sum to 180°; in an n-sided polygon to (n − 2) × 180°.
With parallel lines: alternate angles are equal, corresponding angles are equal, co-interior angles sum to 180°.
Similar triangles have equal angles and proportional sides.

Angle facts

Vertically opposite angles are equal. In an isosceles triangle the base angles are equal, so if the apex is $40°$ , each base angle is $\frac{180 - 40}{2} = 70°$ .

Parallel lines and polygons

A line crossing two parallel lines makes equal alternate ("Z") and corresponding ("F") angles; co-interior ("C") angles add to $180°$ .

The interior angles of a hexagon sum to $(6 - 2) \times 180° = 720°$ , so each angle of a regular hexagon is $120°$ . Exterior angles of any polygon sum to $360°$ .

Similar triangles

If two triangles share their angles, corresponding sides are in the same ratio. When a line parallel to one side cuts a triangle, it creates a smaller similar triangle, a common SAT and GCSE setup.

Worked examples

Example 1 · Regular polygon

Find the size of each interior angle of a regular octagon.

Step 1.Sum $= (8 - 2) \times 180° = 1080°$ .
Step 2.Each $= 1080° \div 8 = 135°$ .

Answer: 135°

Example 2 · Similar triangles

In triangle $ABC$ , $DE$ is parallel to $BC$ with $D$ on $AB$ and $E$ on $AC$ . $AD = 4$ , $AB = 10$ and $BC = 15$ . Find $DE$ .

Step 1.Triangles $ADE$ and $ABC$ are similar with ratio $\frac{4}{10}$ .
Step 2. $DE = 15 \times \frac{4}{10} = 6$ .

Answer: 6

Common mistakes

Confusing co-interior angles (sum 180°) with alternate angles (equal).
Using (n − 2) × 180° for exterior angles; exterior angles always sum to 360°.
Matching the wrong pairs of sides in similar triangles.

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1Angles, parallel lines and similar trianglesEasier

Two angles of a triangle are 48° and 77°. What is the third angle, in degrees?

Question 2Angles, parallel lines and similar trianglesEasier

An isosceles triangle has an apex angle of 40°. What is each base angle?