Angles on a straight line add up to 180°, angles around a point add up to 360°, and vertically opposite angles are equal. The angles in a triangle add up to 180°. With parallel lines, corresponding angles and alternate angles are equal, and co-interior angles add up to 180°. Exam questions usually ask you to give a reason for each step.
Angle facts
- Angles on a straight line add up to 180°
- Angles around a point add up to 360°
- Vertically opposite angles are equal
- Angles in a triangle add up to 180°; in a quadrilateral, 360°
- Parallel lines: corresponding angles equal, alternate angles equal, co-interior angles add to 180°
- Exterior angles of any polygon add up to 360°
Angles on a line and around a point
Angles that sit together on a straight line add up to 180°, and angles that fill the space around a point add up to 360°. To find a missing angle, subtract the known angles from the total: with 115° on a straight line, the other angle is 65°.
Vertically opposite angles and triangles
When two straight lines cross, the angles opposite each other are equal. The angles in any triangle add up to 180°. In an isosceles triangle, the two base angles are equal, so if the angle at the top is 40°, each base angle is (180 − 40) ÷ 2 = 70°. In an equilateral triangle every angle is 60°.
Angles in parallel lines
When a line crosses two parallel lines, three pairs of angles appear. Corresponding angles, in an F shape, are equal. Alternate angles, in a Z shape, are equal. Co-interior angles, in a C or U shape, add up to 180°. Use the correct name as your reason: “alternate angles are equal”, not “Z angles”.
Angles in polygons
The exterior angles of any polygon add up to 360°, so each exterior angle of a regular polygon is 360° divided by the number of sides. The interior angle is 180° minus the exterior angle. The interior angles of an n-sided polygon add up to (n − 2) × 180°, so a pentagon's interior angles total 540°.
Giving reasons
Many questions say “give a reason for each stage of your working”. Write the angle fact you used next to each calculation, using the standard wording, such as “angles in a triangle add up to 180°”. A correct answer without reasons may lose marks.
Worked examples
Example 1. Two angles on a straight line are 115° and x. Find x.
- Angles on a straight line add up to 180°.
- 180 − 115.
Answer: x = 65°.
Example 2. Three angles around a point are 140°, 95° and y. Find y.
- Angles around a point add up to 360°.
- 360 − 140 − 95.
Answer: y = 125°.
Example 3. A triangle has angles 38° and 76°. Find the third angle.
- Angles in a triangle add up to 180°.
- 180 − 38 − 76.
Answer: 66°.
Example 4. Two parallel lines are crossed by a transversal. One angle is 68°. Find the alternate angle and the co-interior angle.
- Alternate angles are equal, so the alternate angle is 68°.
- Co-interior angles add up to 180°, so 180 − 68 = 112°.
Answer: Alternate 68°; co-interior 112°.
Example 5. Work out the size of each interior angle of a regular hexagon.
- Each exterior angle: 360° ÷ 6 = 60°.
- Interior angle: 180° − 60°.
Answer: 120°.
Common mistakes
- Mixing up alternate and co-interior angles. Alternate angles are equal; co-interior angles add up to 180°.
- Using informal reasons. Write “alternate angles are equal”, not “Z angles”.
- Assuming a triangle is isosceles. Only use equal base angles if the sides are marked equal.
- Forgetting the polygon formula. Interior angles of an n-sided polygon total (n − 2) × 180°.
- Measuring instead of calculating. Diagrams are usually not drawn accurately; calculate with angle facts.
How angles are examined
Angle facts are among the first geometry skills tested on Foundation papers for 1MA1 and 4MA1, and Higher questions build on them with polygons and circle theorems. Typical questions:
- find missing angles on lines, around points and in triangles
- use parallel line angle facts, giving reasons
- find interior and exterior angles of regular polygons
- combine several angle facts in one multi-step diagram
The angle rules worksheet has questions with answers, and the GCSE geometry revision engine gives unlimited practice.
Angle facts lead to circle theorems, and are used alongside trigonometry.
Frequently asked questions
What do angles in a triangle add up to?
180°.
What is the difference between alternate and corresponding angles?
Alternate angles form a Z shape and corresponding angles form an F shape; both pairs are equal.
How do you find the interior angle of a regular polygon?
Work out 360° ÷ the number of sides for the exterior angle, then subtract it from 180°.