Angles on Parallel Lines — KS3 Maths

When a straight line (transversal) crosses two parallel lines, it creates pairs of equal angles and pairs of supplementary angles. The three key rules are: alternate angles are equal, corresponding angles are equal, and co-interior angles add to 180°.

What are parallel lines?

Parallel lines are straight lines that are always the same distance apart and never meet. They are marked with small arrows (→ →) on diagrams.

A transversal is a line that crosses two parallel lines, creating eight angles at the two intersection points.

The three angle rules

Rule 1 — Alternate angles (Z-angles)

Alternate angles are found on opposite sides of the transversal, between the parallel lines. They form a "Z" shape (or "N" shape).

Alternate angles are equal.

If angle a is alternate to angle b, then a = b.

Rule 2 — Corresponding angles (F-angles)

Corresponding angles are in the same position at each intersection — both above the parallel line on the same side of the transversal. They form an "F" shape.

Corresponding angles are equal.

If angle c is corresponding to angle d, then c = d.

Rule 3 — Co-interior angles (C-angles or same-side interior angles)

Co-interior angles are on the same side of the transversal, between the parallel lines. They form a "C" shape (or "U" shape).

Co-interior angles add up to 180°.

If angle e and angle f are co-interior, then e + f = 180°.

Summary table

Angle pair | Position | Relationship | Memory aid
Alternate | Opposite sides, between parallel lines | Equal | Z-shape
Corresponding | Same side, same position at each crossing | Equal | F-shape
Co-interior | Same side, between parallel lines | Sum to 180° | C-shape

Worked examples

Example 1 — find the missing angle using alternate angles

Two parallel lines are cut by a transversal. One angle is 65°. Find the alternate angle.

Alternate angles are equal, so the missing angle = 65°.

Reason to write in your working: "Alternate angles are equal (parallel lines)."

Example 2 — find the missing angle using corresponding angles

A transversal crosses two parallel lines. One of the angles created is 110°. Find the corresponding angle.

Corresponding angles are equal, so the missing angle = 110°.

Reason: "Corresponding angles are equal (parallel lines)."

Example 3 — use co-interior angles

Two parallel lines are crossed by a transversal. One co-interior angle is 73°. Find the other co-interior angle.

Co-interior angles sum to 180°, so the missing angle = 180° − 73° = 107°.

Reason: "Co-interior angles add to 180° (parallel lines)."

Example 4 — multi-step problem

In the diagram, lines AB and CD are parallel. The transversal meets AB at angle 125°. Find the acute angle at the second intersection.

The obtuse angle at the first intersection is 125°.

Step 1: The corresponding angle at the second intersection is also 125° (corresponding angles, parallel lines).

Step 2: The angle on a straight line supplementary to 125° = 180° − 125° = 55°

So the acute angle at the second intersection is 55°.

Example 5 — using all three rules together

Lines p and q are parallel. A transversal creates angle x = 48° above line p, on the left side.

Find the four angles at the second intersection (below line q):

Position | Rule used | Angle
Above line q, left | Corresponding to x | 48°
Above line q, right | Supplementary to above | 132°
Below line q, left | Vertically opposite to above-right | 132°
Below line q, right | Vertically opposite to above-left | 48°

Check: all eight angles at both intersections = four pairs of (48°, 132°), summing as expected. ✓

Giving reasons in geometry

At KS3, examiners expect you to state the geometric reason for each step. Use these exact phrases:

"Alternate angles are equal (parallel lines)"

"Corresponding angles are equal (parallel lines)"

"Co-interior angles add to 180° (parallel lines)"

"Angles on a straight line add to 180°"

"Vertically opposite angles are equal"

A numerical answer without a reason loses marks in geometry questions.

Parallel lines in the national curriculum

The DfE's KS3 mathematics programme of study requires pupils to apply the properties of angles at a point, on a straight line, and in parallel lines, and to use these to find missing angles with reasons. The statutory national curriculum for Key Stage 3 and 4 confirms that alternate, corresponding, and co-interior angle rules are assessed on both KS3 and GCSE papers, almost always requiring a stated geometric reason.

Frequently asked questions

How do I identify which rule to use on an angle diagram?

Look at the position of the angle pair relative to the transversal and the parallel lines. If the angles are on opposite sides of the transversal between the parallel lines, it is alternate (equal). If they are in the same position at each intersection, it is corresponding (equal). If they are on the same side between the parallel lines, it is co-interior (sum to 180°).

Are alternate angles always equal even if the lines are not parallel?

No. The alternate angle rule (and the corresponding angle rule) only hold when the lines are parallel. If the lines are not parallel, the angles are not guaranteed to be equal. This is why you must confirm or state that the lines are parallel before applying these rules.

What is the difference between co-interior and supplementary angles?

Supplementary angles are any two angles that add to 180°. Co-interior angles are a specific pair of supplementary angles formed when a transversal crosses parallel lines, positioned on the same side between the parallel lines. All co-interior angles are supplementary, but not all supplementary angle pairs are co-interior angles.

Can I use these rules in a proof?

Yes. Alternate, corresponding, and co-interior angle rules are accepted reasons in formal geometric proof at KS3 and GCSE. State the rule clearly (e.g. "alternate angles are equal since AB ∥ CD") and it counts as a valid line of reasoning in a multi-step proof.

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