GCSE Maths grade 4 basics · topic 5 of 9

Angles: GCSE Maths grade 4 basics

A handful of angle facts answer most angle questions. Learn the facts, then always write the reason next to each step: reasons often carry their own marks.

Practise: 11 questions Read the basics first

In plain English

Angles on a straight line add up to 180°. Angles around a point add up to 360°.
Angles in a triangle add up to 180°; in a quadrilateral, 360°. Vertically opposite angles are equal.
Parallel lines: alternate angles (Z shape) are equal, corresponding angles (F shape) are equal, co-interior angles (C shape) add up to 180°.
Regular polygons: the exterior angles add up to 360°, so each exterior angle is 360° ÷ the number of sides. Interior + exterior = 180°.

Worked example

A triangle has angles of 35° and 90°. Find the third angle.

Angles in a triangle add up to 180°
35° + 90° = 125°
180° − 125° = 55°

Answer: 55°

Most common mistake

WrongThird angle of a triangle with 48° and 67°: 360° − 48° − 67° = 245°

Right180° − 48° − 67° = 65°

360° is for angles around a point and in a quadrilateral. A triangle's angles add up to 180°. If your angle is bigger than the whole triangle could be, check which fact you used.

Practice questions (11)

Try each one on paper first, then open the worked answer and mark yourself. You don't need a calculator for any of these.

Q1.Two angles lie on a straight line. One is 124°. Find the other.
Show worked answer
1. Angles on a straight line add up to 180°
2. 180° − 124° = 56°
Answer: 56°
Q2.Three angles meet at a point: 90°, 130° and x. Find x.
Show worked answer
1. Angles around a point add up to 360°
2. 90° + 130° = 220°
3. x = 360° − 220° = 140°
Answer: 140°
Q3.A triangle has angles of 48° and 67°. Find the third angle.
Show worked answer
1. Angles in a triangle add up to 180°
2. 48° + 67° = 115°
3. 180° − 115° = 65°
Answer: 65°
Q4.An isosceles triangle has one angle of 40° between its two equal sides. Find the size of each of the other two angles.
Show worked answer
1. The other two angles are equal (base angles of an isosceles triangle)
2. 180° − 40° = 140°
3. 140° ÷ 2 = 70°
Answer: 70° each
Q5.Two straight lines cross. One of the angles is 73°. What is the angle vertically opposite it?
Show worked answer
1. Vertically opposite angles are equal
Answer: 73°
Q6.A quadrilateral has angles of 85°, 95°, 110° and x. Find x.
Show worked answer
1. Angles in a quadrilateral add up to 360°
2. 85° + 95° + 110° = 290°
3. x = 360° − 290° = 70°
Answer: 70°
Q7.A line crosses two parallel lines, making a Z shape. One of the angles inside the Z is 58°. What is the other angle inside the Z?
Show worked answer
1. The angles inside a Z shape are alternate angles
2. Alternate angles are equal
Answer: 58°
Q8.A line crosses two parallel lines. Two co-interior angles (a C shape) are 115° and y. Find y.
Show worked answer
1. Co-interior angles add up to 180°
2. y = 180° − 115° = 65°
Answer: 65°
Q9.Find the size of each exterior angle of a regular hexagon (6 sides).
Show worked answer
1. Exterior angles of a polygon add up to 360°
2. 360° ÷ 6 = 60°
Answer: 60°
Q10.Find the size of each interior angle of a regular pentagon (5 sides).
Show worked answer
1. Exterior angle: 360° ÷ 5 = 72°
2. Interior angle: 180° − 72° = 108°
Answer: 108°
Q11.The angles of a triangle are x, 2x and 3x. Find x.
Show worked answer
1. x + 2x + 3x = 180
2. 6x = 180
3. x = 30°
Answer: x = 30°

Can you do these?

Use angles on a straight line and around a point
Find missing angles in triangles and quadrilaterals
Use alternate, corresponding and co-interior angles
Find interior and exterior angles of regular polygons, giving reasons

Tick them off on the printable checklist. Want more? The Core sections of our free geometry notes cover the same maths (they are written for IGCSE, but the basics are the same).