Angles in Polygons & Parallel Lines (Edexcel IGCSE Maths): Revision notes

Section 1

What are the basic angle facts on a line and at a point?

Two rules underpin almost every angle question:

Angles on a straight line add up to $180^{\circ}$ .
Angles around a point (full turn) add up to $360^{\circ}$ .

When two straight lines cross, they form two pairs of vertically opposite angles, which are always equal. Adjacent angles along the line still add to $180^{\circ}$ .

Example: if one angle at a crossing point is $65^{\circ}$ , the angle vertically opposite it is also $65^{\circ}$ , and the two angles next to it are each $180^{\circ} - 65^{\circ} = 115^{\circ}$ .

Key termsvertically opposite anglesangles on a lineangles at a point

Exam tip

Spot vertically opposite angles by looking for an 'X' shape — the two angles that don't touch are equal.

Common mistake

Do not assume all four angles at a crossing are equal — only the two opposite pairs match; adjacent angles are supplementary, not equal, unless the lines are perpendicular.

Section 2

What angle facts apply to parallel lines cut by a transversal?

When a straight line (a transversal) crosses a pair of parallel lines, three special angle relationships appear:

Corresponding angles are equal — they sit in the same position at each intersection (often described as an 'F' shape).
Alternate angles are equal — they sit on opposite sides of the transversal, between the parallel lines (a 'Z' shape).
Co-interior angles (also called allied angles) sum to $180^{\circ}$ — they sit on the same side of the transversal, between the parallel lines (a 'C' or 'U' shape).

These facts only apply because the lines are marked or stated as parallel (usually shown with matching arrow marks).

Key termstransversalcorresponding anglesalternate anglesco-interior angles

Think of it like this

Think of the letters F, Z and C: F for corresponding (equal), Z for alternate (equal), C for co-interior (adds to $180^{\circ}$ ).

Common mistake

Never apply these angle rules unless the lines are confirmed parallel — exam questions sometimes include a distractor pair of lines that only look parallel.

Section 3

How do you find the interior angle sum of a polygon?

For any polygon with $n$ sides, split it into triangles from one vertex. The number of triangles formed is always $n - 2$ , and each triangle's angles sum to $180^{\circ}$ , so:

$\text{sum of interior angles} = (n - 2) \times 180^{\circ}$

For a regular polygon (all sides and angles equal), divide this total by $n$ to find one interior angle:

$\text{interior angle} = \frac{(n - 2) \times 180^{\circ}}{n}$

Example: a regular hexagon ( $n = 6$ ) has an interior angle sum of $4 \times 180^{\circ} = 720^{\circ}$ , so each interior angle is $720^{\circ} \div 6 = 120^{\circ}$ .

Key termspolygoninterior angleregular polygon

Example

A pentagon has $n = 5$ , so interior angle sum $= 3 \times 180^{\circ} = 540^{\circ}$ . In a regular pentagon each interior angle is $540^{\circ} \div 5 = 108^{\circ}$ .

Section 4

How do exterior angles work, and how do they help solve problems?

The exterior angle at any vertex is the angle between a side and the extension of the adjacent side — it is supplementary to the interior angle at that vertex: $\text{interior} + \text{exterior} = 180^{\circ}$ .

The exterior angles of any convex polygon always sum to $360^{\circ}$ , regardless of the number of sides. For a regular polygon:

$\text{exterior angle} = \frac{360^{\circ}}{n}$

This is often the fastest route to a problem: find the exterior angle first, then subtract from $180^{\circ}$ to get the interior angle, or use $n = \frac{360^{\circ}}{\text{exterior angle}}$ to find the number of sides.

Key termsexterior angle

Exam tip

If told a regular polygon has an exterior angle of $24^{\circ}$ , find the number of sides with $n = 360^{\circ} \div 24^{\circ} = 15$ .

Common mistake

Do not confuse the exterior angle sum ( $360^{\circ}$ , always true for any convex polygon) with the interior angle sum ( $(n-2) \times 180^{\circ}$ , which depends on $n$ ).

Section 5

How do you combine these facts to solve multi-step angle problems?

Exam questions often chain several facts together — for example, using parallel line angles to find one angle in a polygon, then applying the interior angle sum to find an unknown side count or missing angle.

A reliable approach:

Mark all known angles on the diagram (or in your working, if none is given).
Identify which rule applies at each step — line/point, parallel lines, or polygon sums.
Write an equation and solve for the unknown, showing each step.
Check your answer is sensible (e.g. interior angles of a convex polygon are less than $180^{\circ}$ ).

Always state which angle fact justifies each step — mark-schemes award marks for correct reasoning, not just the final number.

Exam tip

Name the rule you're using in your working (e.g. 'co-interior angles sum to $180^{\circ}$ ') — this picks up method marks even if the final answer is wrong.

Must Know

Angles on a straight line sum to $180^{\circ}$ ; angles around a point sum to $360^{\circ}$ .
Vertically opposite angles are equal.
With parallel lines: corresponding angles are equal (F), alternate angles are equal (Z), co-interior angles sum to $180^{\circ}$ (C).
Interior angle sum of a polygon $= (n-2) \times 180^{\circ}$ .
Exterior angles of any convex polygon always sum to $360^{\circ}$ ; for a regular polygon, exterior angle $= 360^{\circ} \div n$ .
Interior angle $+$ exterior angle $= 180^{\circ}$ at any vertex.

That's the notes covered.

Carry on to the next subtopic.