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Angles are one of the most fundamental ideas in geometry. For Edexcel GCSE Mathematics (1MA1), you need to be confident with angle facts on lines and at points, angles in triangles and quadrilaterals, and interior and exterior angles of regular and irregular polygons. This lesson covers all of these topics with worked examples and Edexcel-style exam practice.
| Term | Meaning | Example |
|---|---|---|
| Acute angle | An angle less than 90° | 35°, 60° |
| Right angle | An angle of exactly 90° | Corner of a square |
| Obtuse angle | An angle between 90° and 180° | 120°, 150° |
| Reflex angle | An angle between 180° and 360° | 210°, 300° |
| Polygon | A closed 2D shape with straight sides | Triangle, hexagon |
| Regular polygon | A polygon where all sides and all angles are equal | Equilateral triangle, square |
| Interior angle | An angle inside a polygon at a vertex | — |
| Exterior angle | The angle between one side and the extension of the adjacent side | — |
These are the building blocks you must memorise — they are not given on the Edexcel formula sheet.
Angles on a straight line add up to 180°.
Worked Example 1
Two angles on a straight line are x° and 130°. Find x.
x + 130 = 180 x = 180 - 130 = 50°
Angles around a point add up to 360°.
Worked Example 2
Three angles around a point are 140°, 85° and y°. Find y.
140 + 85 + y = 360 225 + y = 360 y = 135°
When two straight lines cross, the opposite angles are equal.
Worked Example 3
Two straight lines cross. One of the angles is 72°. State the sizes of the other three angles.
The vertically opposite angle = 72°. The other pair of vertically opposite angles = 180 - 72 = 108° each.
The angles in any triangle add up to 180°.
| Triangle | Properties |
|---|---|
| Equilateral | All sides equal, all angles 60° |
| Isosceles | Two sides equal, two base angles equal |
| Scalene | No sides or angles equal |
| Right-angled | One angle of 90° |
Worked Example 4
An isosceles triangle has an angle of 40° between the two equal sides. Find the base angles.
Let each base angle = b. 40 + b + b = 180 40 + 2b = 180 2b = 140 b = 70°
The angles in any quadrilateral add up to 360°.
Worked Example 5
Three angles of a quadrilateral are 90°, 85° and 110°. Find the fourth angle.
90 + 85 + 110 + d = 360 285 + d = 360 d = 75°
An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Worked Example 6
In triangle PQR, angle P = 55° and angle Q = 70°. Side QR is extended to point S. Find the exterior angle PRS.
Exterior angle PRS = P + Q = 55 + 70 = 125°
For a polygon with n sides:
Sum of interior angles = (n - 2) x 180°
| Polygon | Sides (n) | Sum of interior angles |
|---|---|---|
| Triangle | 3 | (3 - 2) x 180 = 180° |
| Quadrilateral | 4 | (4 - 2) x 180 = 360° |
| Pentagon | 5 | (5 - 2) x 180 = 540° |
| Hexagon | 6 | (6 - 2) x 180 = 720° |
| Octagon | 8 | (8 - 2) x 180 = 1080° |
| Decagon | 10 | (10 - 2) x 180 = 1440° |
Each interior angle = (n - 2) x 180° / n
Worked Example 7
Find the size of each interior angle of a regular octagon.
Sum = (8 - 2) x 180 = 1080° Each angle = 1080 / 8 = 135°
The exterior angles of any convex polygon always add up to 360°.
For a regular polygon: Each exterior angle = 360° / n
Important: Interior angle + Exterior angle = 180° (they form a straight line).
Worked Example 8
Each exterior angle of a regular polygon is 40°. How many sides does it have?
n = 360 / 40 = 9 sides (nonagon)
Worked Example 9
Each interior angle of a regular polygon is 156°. Find the number of sides.
Exterior angle = 180 - 156 = 24° n = 360 / 24 = 15 sides
When a transversal crosses two parallel lines, the following angle relationships hold:
| Type | Rule | How to spot |
|---|---|---|
| Alternate angles | Equal | Z-shape (or reversed Z) |
| Corresponding angles | Equal | F-shape (or reversed F) |
| Co-interior (allied) angles | Add up to 180° | C-shape or U-shape |
Worked Example 10
A transversal crosses two parallel lines. One of the alternate angles is 64°. Find the co-interior angle on the same side.
Alternate angle = 64°, so the other alternate angle = 64°. Co-interior angle = 180 - 64 = 116°