Use angle facts on straight lines, around points, in parallel lines and in polygons. The diagrams below show the given information; worked answers stay closed until you choose to view them.
Foundation and Higher · Useful for students strengthening core skills before mixed GCSE practice.
Core pathway. Core GCSE methods are useful across Foundation and Higher. Start with the core examples and questions.
Higher extension. More demanding selections are marked Higher extension: practice question 8. Use these after the core work, following your teacher’s guidance.
3 worked examples and 8 original practice questions · Allow 25–40 minutes · Read online or print. Higher extensions are labelled. No account or email needed.
Printing includes the method, examples and space for working, followed by a separate answer section.
A useful approach
Identify a relevant fact: angles on a straight line total 180°, and angles around a point total 360°.
Check that lines are stated to be parallel before using corresponding or alternate angle facts.
Write a reason beside each calculation. Use the polygon's number of sides to choose its angle-sum calculation.
A mistake to watch for
A sketch is not a reliable protractor. Use the stated measurements and geometric facts, even when an angle looks different on the screen.
Learn each method, then practise it
Read each line of working and explain why it follows from the previous line.
Straight lines and points
Angles on a straight line total 180°. Around a point they total 360°. Write the equation before solving for an unknown.
Worked example 1
Angles · Core GCSE skill · No calculator · 3 marks
AOB is a straight line. The point C lies above the line.
Angle AOC = 5x° and angle COB = (4x − 9)°.
Work out the value of x. Give a reason for each stage of your working.
Diagram for Angles. The question states the required measurements and relationships.Read a text description of this diagram
Given diagram. AOB is a straight line. The point C lies above the line. Angle AOC = 5 x degrees and angle COB = (4 x minus 9) degrees. Work out the value of x . Give a reason for each stage of your working. Use the labelled information; do not estimate measurements from the sketch.
Angles on a straight line add up to 180°, so 5x + 4x − 9 = 180.
9x − 9 = 180, so 9x = 189.
x = 189 ÷ 9 = 21.
Check: 5 × 21 = 105 and 4 × 21 − 9 = 75, and 105 + 75 = 180.
Use alternate or corresponding angles only when the lines are stated to be parallel. Write a reason beside each step. Co-interior angles between parallel lines total 180°.
Worked example 2
Angles in Parallel Lines · Core GCSE skill · No calculator · 4 marks
AB and CD are parallel straight lines.
A straight line crosses AB at E and crosses CD at F. On line AB, A is to the left of E and B is to the right of E. On line CD, C is to the left of F and D is to the right of F.
Angle AEF = 62°.
Work out the size of angle CFE.
Give a reason for each stage of your working.
Diagram for Angles in Parallel Lines. The question states the required measurements and relationships.Read a text description of this diagram
Given diagram. AB and CD are parallel straight lines. A straight line crosses AB at E and crosses CD at F. On line AB, A is to the left of E and B is to the right of E. On line CD, C is to the left of F and D is to the right of F. Angle AEF = 62 degrees. Work out the size of angle CFE. Give a reason for each stage of your working. Use the labelled information; do not estimate measurements from the sketch.
Alternate angles lie between parallel lines on opposite sides of the crossing line. Co-interior angles lie between the parallel lines on the same side of the crossing line.
Angle EFD = 62° because angle AEF and angle EFD are alternate angles, and alternate angles are equal
Angle CFE = 180° − 62° = 118° because angles on a straight line add up to 180°
(Alternatively: angle AEF and angle CFE are co-interior angles, so angle CFE = 180° − 62° = 118°)
The exterior angles total 360°. For a regular polygon, divide by the number of sides; each interior angle is supplementary to its exterior angle.
Worked example 3
Angles in Polygons · Core GCSE skill · No calculator · 2 marks
Each exterior angle of a regular polygon is 30°.
Work out the number of sides of the polygon.
The exterior angles of any polygon add up to 360°
Number of sides = 360 ÷ 30 = 12
Walking once around a convex polygon turns you through one complete turn, 360°. In a regular polygon each turn is equal, so divide 360° by the exterior angle.
Write your working on paper. Marks indicate how much working to show; these questions are self-marked and do not change saved practice results. Use squared paper for drawing questions.
Question 1
Angles · Core GCSE skill · No calculator · 2 marks
Two angles fit together on a straight line. One of the angles is 64° and the other angle is y°.
Angles in Parallel Lines · Core GCSE skill · Calculator allowed · 4 marks
PQ and RS are parallel straight lines.
A straight line crosses PQ at E and crosses RS at F. On line PQ, P is to the left of E and Q is to the right of E. On line RS, R is to the left of F and S is to the right of F.
Angle QEF = 74°.
Work out the size of angle RFE. Give a reason for your answer.
Work out the size of angle EFS. Give a reason for your answer.
Diagram for Angles in Parallel Lines. The question states the required measurements and relationships.Read a text description of this diagram
Given diagram. PQ and RS are parallel straight lines. A straight line crosses PQ at E and crosses RS at F. On line PQ, P is to the left of E and Q is to the right of E. On line RS, R is to the left of F and S is to the right of F. Angle QEF = 74 degrees. Work out the size of angle RFE. Give a reason for your answer. Work out the size of angle EFS. Give a reason for your answer. Use the labelled information; do not estimate measurements from the sketch.
Angles in Parallel Lines · Core GCSE skill · Calculator allowed · 4 marks
AB and CD are parallel straight lines.
A straight line GEFH crosses AB at E and crosses CD at F, where G is above the line AB and H is below the line CD. On line AB, A is to the left of E and B is to the right of E. On line CD, C is to the left of F and D is to the right of F.
Angle GEB = 63°.
Work out the size of angle EFD. Give a reason for your answer.
Work out the size of angle CFH. Give a reason for your answer.
Diagram for Angles in Parallel Lines. The question states the required measurements and relationships.Read a text description of this diagram
Given diagram. AB and CD are parallel straight lines. A straight line GEFH crosses AB at E and crosses CD at F, where G is above the line AB and H is below the line CD. On line AB, A is to the left of E and B is to the right of E. On line CD, C is to the left of F and D is to the right of F. Angle GEB = 63 degrees. Work out the size of angle EFD. Give a reason for your answer. Work out the size of angle CFH. Give a reason for your answer. Use the labelled information; do not estimate measurements from the sketch.
Angles in Parallel Lines · Core GCSE skill · Calculator allowed · 4 marks
DE and FG are parallel straight lines.
Triangle ABC is drawn so that A lies on DE, and B and C both lie on FG.
D is to the left of A and E is to the right of A.
Angle DAB = 48° and angle ACB = 63°.
Work out the size of angle BAC.
Give a reason for each stage of your working.
Diagram for Angles in Parallel Lines. The question states the required measurements and relationships.Read a text description of this diagram
Given diagram. DE and FG are parallel straight lines. Triangle ABC is drawn so that A lies on DE, and B and C both lie on FG. D is to the left of A and E is to the right of A. Angle DAB = 48 degrees and angle ACB = 63 degrees. Work out the size of angle BAC. Give a reason for each stage of your working. Use the labelled information; do not estimate measurements from the sketch.
Compare the reasoning as well as the final answer. Another correct method is valid. If a step is unclear, revisit an example before trying a similar question.
Answer 1
Show answer and working for question 1
Angles on a straight line add up to 180°.
y = 180 − 64 = 116.
Answer: (a) 116 (b) angles on a straight line add up to 180 degrees
Draw diagonals from one vertex of a convex decagon: they divide it into 10 − 2 = 8 triangles. Each triangle contributes 180°, explaining the interior-angle sum formula.
A decagon has 10 sides, so the sum of its interior angles is (10 − 2) × 180° = 1440°
Each interior angle = 1440 ÷ 10 = 144°
(Alternatively: each exterior angle = 360 ÷ 10 = 36°, so each interior angle = 180 − 36 = 144°)
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Use this in class
Display one diagram and ask pupils to name the usable facts before finding a missing angle. Zoom keeps small labels visible without adding solution values.
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