Angles and Polygons

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.

Key Vocabulary

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	—

Basic Angle Facts

These are the building blocks you must memorise — they are not given on the Edexcel formula sheet.

Angles on a Straight Line

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 at a Point

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°

Vertically Opposite Angles

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.

Angles in Triangles

The angles in any triangle add up to 180°.

Special Triangles

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°

Angles in Quadrilaterals

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°

Exterior Angles of a Triangle

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°

Interior and Exterior Angles of Polygons

Sum of Interior Angles

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 of a Regular Polygon

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°

Exterior Angles

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

Angles in Parallel Lines

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°

Common Mistakes and Misconceptions

Forgetting that angle sums work for irregular polygons too. The formula (n - 2) x 180° applies to all polygons, not just regular ones.
Confusing interior and exterior angles. Remember: they add up to 180° at each vertex.
Using the wrong angle fact on parallel lines. Check the shape formed: Z = alternate, F = corresponding, C/U = co-interior.
Writing "angles on a straight line = 360°." No — angles on a straight line = 180°; angles at a point = 360°.
Not giving reasons in exam answers. Edexcel requires you to state the angle fact used (e.g. "co-interior angles sum to 180°").

Edexcel Exam Tips

Always state which angle fact you are using — marks are awarded for the reasoning, not just the numerical answer.
"Give reasons for your answer" means you must name the geometric property (e.g. "vertically opposite angles are equal").
Polygon interior angle sums are NOT on the Edexcel formula sheet — you must memorise (n - 2) x 180°.
If a question involves parallel lines, look for arrows on the diagram indicating which lines are parallel.

Practice Problems

Two angles on a straight line are 3x° and (x + 20)°. Find x. Answer: 3x + x + 20 = 180, 4x = 160, x = 40.
The angles of a triangle are (2y + 10)°, (3y)° and (y + 20)°. Find y and each angle. Answer: 6y + 30 = 180, y = 25; angles = 60°, 75°, 45°.
Find the sum of the interior angles of a heptagon (7 sides). Answer: (7 - 2) x 180 = 900°.
Each interior angle of a regular polygon is 144°. How many sides? Answer: Exterior = 36°, n = 360/36 = 10 sides.
In the diagram, lines AB and CD are parallel. A transversal makes an angle of 52° with AB. Find the co-interior angle at CD. Answer: 180 - 52 = 128°.

Summary

Angles on a straight line = 180°; at a point = 360°; vertically opposite angles are equal.
Angles in a triangle = 180°; in a quadrilateral = 360°.
Sum of interior angles of an n-sided polygon = (n - 2) x 180°.
Exterior angles of any convex polygon = 360°.
Interior + exterior angle = 180° at each vertex.
Parallel line angles: alternate (equal), corresponding (equal), co-interior (sum to 180°).
Always give geometric reasons in your exam answers.