Almost every angles question relies on a small set of core facts. Learning these thoroughly makes even the hardest polygon and parallel-line problems straightforward.
Angles on a straight line sum to 180°
Angles around a point sum to 360°
Vertically opposite angles (formed where two lines cross) are equal
Worked Example 1
Angles ABC and CBD lie on the straight line AD. Angle ABC = 3x° and angle CBD = (x + 40)°. Find the value of x.
1
Angles on a straight line sum to 180°, so: 3x + (x + 40) = 180
2
Simplify: 4x + 40 = 180
3
Solve: 4x = 140, so x = 35
Answerx = 35
Angles ABC and CBD sit on the straight line AD, so together they make 180°.
Angles in parallel lines
When a straight line (a transversal) crosses two parallel lines, it creates three important angle pairs. Look for the shape each pair makes to identify which rule applies.
Corresponding angles (the "F" shape)
Corresponding angles sit in matching positions at each intersection — trace the transversal down from the top line to the bottom line, and both marked angles are on the same side. The two highlighted segments below trace out the "F" shape.
Lines l₁ and l₂ are parallel (shown by the matching arrow marks). The transversal t crosses both — the two angles marked a° are corresponding angles and are always equal.
Alternate angles (the "Z" shape)
Alternate angles sit between the two parallel lines, on opposite sides of the transversal. The highlighted path below traces the "Z" shape connecting them.
The two angles marked b° lie between l₁ and l₂, on opposite sides of the transversal t — this makes them alternate angles, which are always equal.
Co-interior angles (the "C" shape)
Co-interior (also called "allied") angles sit between the two parallel lines, on the same side of the transversal. Unlike the other two pairs, these are not equal — they sum to 180°.
The angles marked c° and d° are both on the right of the transversal, between l₁ and l₂ — these co-interior angles always satisfy c + d = 180°.
Summary
Angle pair
Memory shape
Rule
Corresponding
"F" shape
Equal
Alternate
"Z" shape
Equal
Co-interior (Allied)
"C" shape
Sum to 180°
Worked Example 2
Two parallel lines are cut by a transversal. One angle is 118°. Find the alternate angle, marked y°.
1
Identify the angle pair: the marked angles form a "Z" shape, so they are alternate angles.
2
Alternate angles are equal, so y = 118°.
Answery = 118°
Angles in triangles and quadrilaterals
Angles in a triangle always sum to 180°. Angles in any quadrilateral always sum to 360° — you can prove this by splitting a quadrilateral into two triangles.
Worked Example 3
A triangle has angles of 52°, 67° and z°. Find the value of z.
1
Angles in a triangle sum to 180°: 52 + 67 + z = 180
2
Simplify: 119 + z = 180
3
Solve: z = 61
Answerz = 61°
Interior and exterior angles of polygons
For any polygon with n sides, the sum of the interior angles is (n − 2) × 180°. The exterior angles of any polygon always sum to 360°. For a regular polygon (all sides and angles equal), divide these sums by n to find each individual angle.
Worked Example 4
Find the size of each interior angle of a regular hexagon (n = 6).
The hexagon is regular, so divide equally between the 6 angles: 720° ÷ 6 = 120°
Answer120°
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Two angles lie on a straight line. One angle is 65°. Find the other angle.Foundation
Q2Three angles around a point are 140°, 95° and w°. Find w.Foundation
Q3Two straight lines cross. One angle is 72°. Find the vertically opposite angle.Foundation
Diagram for Question 3 (not to scale).
Q4A triangle has angles of 48° and 63°. Find the third angle.Foundation
Q5Find the sum of the interior angles of a quadrilateral.Foundation
Higher (Grade 5–7)
Q6Two parallel lines are cut by a transversal. One angle is 105°. Find the corresponding angle.Higher
Diagram for Question 6 (not to scale).
Q7Angles of (2x)° and (x + 30)° lie on a straight line. Find x.Higher
Q8A quadrilateral has angles of 90°, 85°, 110° and y°. Find y.Higher
Q9Find the sum of the interior angles of a regular pentagon (n = 5).Higher
Q10Find the size of each exterior angle of a regular octagon (n = 8).Higher
Higher — Hard (Grade 8–9)
Q11Two parallel lines are cut by a transversal. One angle is (3x + 10)° and its co-interior angle is (2x + 30)°. Find x.Grade 8–9
Q12The exterior angle of a regular polygon is 24°. Find the number of sides.Grade 8–9
Q13A triangle has angles of (2x)°, (3x − 10)° and (x + 40)°. Find x and the size of each angle.Grade 8–9
Q14An interior angle of a regular polygon is 156°. Find the number of sides.Grade 8–9
Q15In triangle ABC, angle A = 90°, angle B = (4x + 10)° and angle C = (2x − 4)°. Find x.Grade 8–9
Answers
Foundation (Q1–Q5)
Q1115°(180 − 65)
Q2125°(360 − 140 − 95)
Q372°
Q469°(180 − 48 − 63)
Q5360°
Higher (Q6–Q10)
Q6105°
Q7x = 50(2x + x + 30 = 180)
Q875°(360 − 90 − 85 − 110)
Q9540°((5−2) × 180)
Q1045°(360 ÷ 8)
Higher — Hard (Q11–Q15)
Q11x = 28(3x+10 + 2x+30 = 180)
Q1215 sides(360 ÷ 24)
Q13x = 25; angles are 50°, 65° and 65°(2x + 3x−10 + x+40 = 180)
Q1415 sides(exterior = 180−156 = 24°, then 360÷24)
Q15x = 14(90 + 4x+10 + 2x−4 = 180)
Common mistakes
Common Mistake 1
Confusing alternate and corresponding angles
"Z" angles (alternate) and "F" angles (corresponding) are both equal, but they sit in different positions relative to the transversal. Mixing them up with "C" angles (co-interior, which sum to 180°) is one of the most common errors in this topic.
Common Mistake 2
Using 180° instead of 360° for angles around a point
Angles on a straight line sum to 180°, but angles fully around a point sum to 360°. Always check whether the angles form a full turn or just a straight line before choosing which rule to use.
Common Mistake 3
Dividing the angle sum by n for an irregular polygon
(n − 2) × 180° only gives the sum of interior angles. Dividing this sum by n to find "each" angle only works if the polygon is regular — for an irregular polygon, the angles can all be different sizes.
Common Mistake 4
Not stating the angle rule used
GCSE mark schemes award marks for stating which angle fact was used — for example "co-interior angles sum to 180°" — not just for writing down the final number.
Exam tips
💡 Exam Tip 1
Always state the angle rule you're using
Write the reason alongside your working, e.g. "angles on a straight line sum to 180°" — this secures method marks even if you make a later arithmetic slip.
💡 Exam Tip 2
Look for the "Z", "F" and "C" shapes
In parallel line diagrams, tracing the shape made by the two marked angles instantly tells you whether they're alternate, corresponding or co-interior.
💡 Exam Tip 3
Check for "regular" before dividing by n
Only divide an interior or exterior angle sum by the number of sides if the question confirms the polygon is regular — otherwise you can only find the total, not an individual angle.
💡 Exam Tip 4
Sanity-check your final answer
Add your angles back together to confirm they match the expected total — 180° for a triangle or straight line, 360° for a quadrilateral or full turn.
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