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This lesson covers the fundamental angle facts and properties you need for AQA GCSE Mathematics. You will learn about angles on straight lines, around a point, and vertically opposite angles, then move on to angles formed by parallel lines and a transversal. Finally, you will explore the interior and exterior angles of polygons. These topics appear frequently on both Foundation and Higher tier papers.
Before tackling more advanced problems, you must be confident with the core angle rules.
| Rule | Statement | Diagram Hint |
|---|---|---|
| Angles on a straight line | Angles on a straight line add up to 180 degrees | A straight line with angles above it |
| Angles around a point | Angles around a point add up to 360 degrees | Angles radiating from a single point |
| Vertically opposite angles | Vertically opposite angles are equal | Two intersecting lines forming an X |
| Angles in a triangle | Angles in a triangle add up to 180 degrees | Any triangle |
| Angles in a quadrilateral | Angles in a quadrilateral add up to 360 degrees | Any four-sided shape |
Two angles on a straight line are x and 130 degrees. Find x.
Solution: Angles on a straight line sum to 180 degrees. x + 130 = 180 x = 180 - 130 x = 50 degrees
Three angles meet at a point: 90 degrees, 145 degrees and y. Find y.
Solution: Angles around a point sum to 360 degrees. 90 + 145 + y = 360 235 + y = 360 y = 360 - 235 y = 125 degrees
Exam Tip: Always state the angle rule you are using when showing your working. Simply writing the calculation without naming the rule will cost you marks on "give a reason" questions.
When a straight line (called a transversal) crosses two parallel lines, several angle relationships are formed.
graph TD
A[Transversal crosses parallel lines] --> B[Alternate angles]
A --> C[Corresponding angles]
A --> D[Co-interior angles]
B --> B1[Equal — Z shape]
C --> C1[Equal — F shape]
D --> D1[Sum to 180 degrees — C or U shape]
| Angle Type | Relationship | Memory Aid |
|---|---|---|
| Alternate angles | Equal | Look for a Z shape (or reversed Z) |
| Corresponding angles | Equal | Look for an F shape (or rotated F) |
| Co-interior angles (also called allied or same-side interior) | Add up to 180 degrees | Look for a C or U shape |
A transversal crosses two parallel lines. One angle is 72 degrees. Find its alternate angle and the co-interior angle on the same side.
Solution:
In a diagram with parallel lines, angle a = 3x + 10 and its corresponding angle b = 5x - 30. Find x.
Solution: Corresponding angles are equal: 3x + 10 = 5x - 30 10 + 30 = 5x - 3x 40 = 2x x = 20
Exam Tip: Questions often combine parallel-line rules with other angle facts. For example, you might need to use alternate angles AND angles in a triangle in the same question. Look for the chain of reasoning.
| Type | Properties |
|---|---|
| Acute-angled | All angles less than 90 degrees |
| Right-angled | One angle is exactly 90 degrees |
| Obtuse-angled | One angle is greater than 90 degrees |
| Type | Properties |
|---|---|
| Equilateral | All three sides equal; all angles are 60 degrees |
| Isosceles | Two sides equal; two base angles equal |
| Scalene | All three sides different; all three angles different |
An isosceles triangle has one angle of 40 degrees at the apex. Find the base angles.
Solution: Let each base angle be b. 40 + b + b = 180 40 + 2b = 180 2b = 140 b = 70 degrees
Exam Tip: In isosceles triangle questions, always identify which angle is the "odd one out." If the given angle is a base angle, then the other base angle is the same. If the given angle is the apex, subtract from 180 and halve the remainder.
The sum of the interior angles of any polygon depends on the number of sides.
Formula: Sum of interior angles = (n - 2) x 180 degrees, where n is the number of sides.
| Polygon | Sides (n) | Sum of Interior Angles |
|---|---|---|
| Triangle | 3 | (3-2) x 180 = 180 degrees |
| Quadrilateral | 4 | (4-2) x 180 = 360 degrees |
| Pentagon | 5 | (5-2) x 180 = 540 degrees |
| Hexagon | 6 | (6-2) x 180 = 720 degrees |
| Heptagon | 7 | (7-2) x 180 = 900 degrees |
| Octagon | 8 | (8-2) x 180 = 1080 degrees |
| Decagon | 10 | (10-2) x 180 = 1440 degrees |
For a regular polygon (all sides and angles equal), each interior angle = (n - 2) x 180 / n.
Find the size of each interior angle of a regular nonagon (9 sides).
Solution: Sum of interior angles = (9 - 2) x 180 = 7 x 180 = 1260 degrees Each interior angle = 1260 / 9 = 140 degrees
The exterior angles of any polygon always sum to 360 degrees.
For a regular polygon, each exterior angle = 360 / n.
Also note: interior angle + exterior angle = 180 degrees (they form a straight line).
The exterior angle of a regular polygon is 40 degrees. How many sides does it have?
Solution: n = 360 / exterior angle = 360 / 40 = 9 sides (a nonagon)
A regular polygon has interior angles of 156 degrees. Find the number of sides.
Solution: Exterior angle = 180 - 156 = 24 degrees n = 360 / 24 = 15 sides
Exam Tip: If you are asked to find the number of sides from an interior angle, always convert to the exterior angle first, then divide 360 by it. This is quicker and less error-prone than using the interior angle formula directly.
Exam Tip: When answering multi-step angle problems in the exam, present your working as a chain of reasoning. State the angle rule at each step — for example, "Angle ABD = 72 degrees (alternate angles are equal)." This is essential for full marks on proof-style questions.