G3: Angles
Apply angle properties at a point, on a straight line, vertically opposite; parallel lines; angles in triangles and polygons
📋 Key Concepts
Basic Angle Facts:
- Angles at a point sum to 360°
- Angles on a straight line sum to 180°
- Vertically opposite angles are equal
📝 Angles at a Point
Rule: Angles around a point add up to 360°.
a + b + c + d = 360°
Example 1
Three angles at a point are 85°, 120° and x. Find x.
Solution:
85° + 120° + x = 360°
205° + x = 360°
x = 360° - 205° = 155°
📝 Angles on a Straight Line
Rule: Angles on a straight line add up to 180°.
Example 2
Two angles on a straight line are 67° and x. Find x.
Solution:
67° + x = 180°
x = 180° - 67° = 113°
📝 Vertically Opposite Angles
Rule: When two lines cross, the angles opposite each other are equal.
Example 3
Two lines intersect. If one angle is 48°, find the other three angles.
Solution:
The vertically opposite angle is also 48°.
The other two angles: 180° - 48° = 132° each (angles on a straight line)
Angles are: 48°, 132°, 48°, 132°
📝 Parallel Lines
When a transversal crosses parallel lines:
- Corresponding angles are equal (F-shape)
- Alternate angles are equal (Z-shape)
- Co-interior angles sum to 180° (C-shape)
Example 4
Parallel lines are cut by a transversal. If an alternate angle is 72°, find the corresponding angle.
Solution:
Alternate angles = 72°
Corresponding angles are also 72° (they are equal to alternate angles)
Example 5
Find angle x if co-interior angles are 72° and x.
Solution:
Co-interior angles sum to 180°
72° + x = 180°
x = 108°
📝 Angles in Triangles
Rule: Angles in a triangle add up to 180°.
Example 6
Two angles in a triangle are 43° and 58°. Find the third angle.
Solution:
43° + 58° + x = 180°
101° + x = 180°
x = 79°
📝 Angles in Polygons
Sum of interior angles of an n-sided polygon:
Sum of interior angles = (n - 2) × 180°
Interior angle of a regular polygon:
Each interior angle = [(n - 2) × 180°] ÷ n
Sum of exterior angles: Always 360° for any polygon.
Example 7
Find the sum of interior angles of a hexagon.
Solution:
Sum = (6 - 2) × 180° = 4 × 180° = 720°
Example 8
Find the interior angle of a regular octagon.
Solution:
Each interior angle = [(8 - 2) × 180°] ÷ 8
= (6 × 180°) ÷ 8 = 1080° ÷ 8 = 135°
🧮 Worked Examples — Exam Style
Angle reasoning chains one fact to the next: point (360°), line (180°), vertically opposite (equal), parallel (Z/F rules), triangle (180°), polygon ((n − 2) × 180°).
Two angles on a straight line are (3x)° and (x + 20)°. Find x.
3x + x + 20 = 180 → 4x = 160 → x = 40 (angles 120° and 60°)
M1 for summing to 180, A1 for x = 40. 'Angles on a straight line add to 180°' is the reason to quote.
A regular polygon has exterior angles of 24°. How many sides does it have?
Exterior angles always total 360° → 360 ÷ 24 = 15 sides
M1 for 360 ÷ 24, A1 for 15. The exterior-angle total (360°) is the fastest polygon tool — memorise it.
Exam wording to expect: "Work out the size of angle …" (give reasons), "Find x".
❓ Practice Questions
Q1: Angles at a point are 98°, 112° and x. Find x.
Q2: Two angles on a straight line are 74° and x. Find x.
Q3: If vertically opposite angles are both 53°, what are the other two angles?
Q4: Find the sum of interior angles of a decagon (10 sides).
Q5: Find the exterior angle of a regular pentagon.
Q6: Find the interior angle sum of a regular hexagon, and each interior angle.
Q7: In a triangle, angles are 2x°, 3x° and 4x°. Find x and the largest angle.
✅ Answers
- 150°
- 106°
- 127° each
- (10 - 2) × 180° = 1440°
- 360° ÷ 5 = 72°
🧮 Worked Examples — Harder Exam Style
AB and CD are parallel lines. PQ cuts them. One alternate angle is 65°. Find all four angles at the other intersection, with reasons.
At the first intersection the four angles are 65°, 115°, 65°, 115° (supplementary + vertically opposite).
At the second intersection: corresponding to 65° is 65° (F rule); its supplement is 115° (allied/co-interior); vertically opposite pairs repeat: 65°, 115°, 65°, 115°
M1 for corresponding/co-interior use, A1 for all four. Quote a named rule for every line of working — that is how the 9-marker scores.
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 98 + 112 = 210 → x = 360 − 210 = 150° (angles at a point sum to 360°)
Q2. 180 − 74 = 106°
Q3. Vertically opposite pairs: 53° each, the other pair = (180 − 53) = 127° each
Q4. (10 − 2) × 180 = 1440°
Q5. 360 ÷ 5 = 72°
Q6. Sum = (6 − 2) × 180 = 720°; each = 720 ÷ 6 = 120°
Q7. 9x = 180 → x = 20; largest = 4x = 80°
📌 Key Facts to Memorise
- Point: 360° · Straight line: 180° · Vertically opposite: equal
- Parallel lines: alternate (Z) equal · corresponding (F) equal · co-interior (C) sum 180°
- Triangle: 180° · Quadrilateral: 360° · n-gon: (n − 2) × 180°
- Exterior angles of ANY polygon total 360°; regular → each = 360 ÷ n
- Base angles of isosceles are equal; equilateral = 60° each
🗺️ In the Exam
Angle chains: 2–5 marks on Paper 1 (non-calculator). Every numeric answer needs its REASON — a correct number with no reason loses half the marks.
Links to: quadrilaterals (G4), polygons and symmetry (G), circle theorems (G10).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always start by finding angles you can work out directly
- Look for parallel line markers (arrows on the lines)
- Mark angles on the diagram as you find them
- Check: interior + exterior angle = 180°
- Remember: all exterior angles add to 360°
🧠 Problem-Solving Strategies
Problem-Solving
Use angle facts systematically: angles on a straight line = 180°, angles at a point = 360°, angles in a triangle = 180°, angles in a quadrilateral = 360°. For parallel lines: corresponding angles are equal, alternate angles are equal, co-interior angles sum to 180°. Always start from known angles and work towards the unknown.
Multi-Step Problem
In the diagram, AB is parallel to CD. A transversal crosses both lines. One corresponding angle is (3x + 10)° and the other is (5x − 20)°. Find x and the size of each angle.
Solution: Corresponding angles are equal: 3x + 10 = 5x − 20. 30 = 2x, x = 15. Each angle = 3(15) + 10 = 55°.
⚠️ Common Errors
Watch Out!
1. Wrong: Saying co-interior (allied) angles are equal Correct: Co-interior angles between parallel lines are supplementary (sum to 180°), not equal. Corresponding and alternate angles are the ones that are equal.
2. Wrong: Adding all angles in a triangle to get 360° Correct: Angles in a triangle sum to 180°. Angles in a quadrilateral sum to 360°.
3. Wrong: Confusing alternate angles with corresponding angles Correct: Alternate angles are on opposite sides of the transversal, inside the parallel lines (Z-shape). Corresponding angles are on the same side of the transversal, one inside and one outside (F-shape).
✍️ 6-Mark Exam Question
Extended Answer
6 marks: In the diagram, PQ is parallel to RS. Line MN crosses both parallel lines. ∠PMT = 3x° and ∠MNS = (x + 40)°, where T is on the same side as S. Point O lies on MN between the parallel lines, and ∠POM = 2x°. (a) State the angle relationship between ∠PMT and ∠MNS. (b) Find x. (c) Find ∠POM. Is it equal to ∠MNS? Explain why or why not.
(a) ∠PMT and ∠MNS are co-interior angles (same side of transversal, inside parallel lines), so they sum to 180°.
(b) 3x + (x + 40) = 180. 4x + 40 = 180. 4x = 140. x = 35.
(c) ∠POM = 2 × 35 = 70°. ∠MNS = 35 + 40 = 75°. They are NOT equal. ∠POM is not necessarily an alternate or corresponding angle to ∠MNS since O is an interior point, not on the parallel lines.
Mark scheme: A1 co-interior, M1 setting up equation, A1 x = 35, A1 ∠POM = 70°, M1 comparison, A1 explanation
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A student measures the three angles of a triangle as 58°, 73° and 50°.
(a) Show that these measurements contain an error.
(b) The student's protractor has a precision of ±1°. Could the triangle actually have a right angle? Explain.
(c) Another triangle has angles in the ratio 2:3:4. Find the angles and classify the triangle.
Answers: (a) 58 + 73 + 50 = 181°, but angles in a triangle sum to 180°. The measurements are inconsistent. (b) If one angle is actually 90° (measured as 73+1=74 or 58+1=59), the other two would need to sum to 90°. 58+50=108 (too high), 73+50=123 (too high). With ±1° precision, no combination gives 90° for one angle, so it is unlikely to be right-angled. (c) Total parts = 9. One part = 180/9 = 20°. Angles = 40°, 60°, 80°. All angles are different and all are acute, so it is an acute scalene triangle.
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