KS3 · Maths

Angles on parallel lines (corresponding, alternate, co-interior)

Your eye can't tell 'nearly parallel' from 'parallel'. The angles made by a crossing line can — and once the lines are parallel, one angle unlocks the rest.

Try to break it

Tilt one line and watch the angle facts fall apart
67°67°67°113°ABPCD

d, at D 67°. Corresponding to d 67°. Alternate to d 67°. Co-interior with d 113°. Relationship: While CD is parallel to AB: d equals its corresponding angle, d equals its alternate angle, and d + its co-interior angle = 180°. Tilt CD and all three stop being true.

d, at D67°Corresponding to d67°Alternate to d67°Co-interior with d113°Drag D up or down the slanted line

While CD is parallel to AB: d equals its corresponding angle, d equals its alternate angle, and d + its co-interior angle = 180°. Tilt CD and all three stop being true.

The slanted line is a transversal: a line that crosses other lines at different points. CD starts parallel to AB — on paper you would show that with matching arrow marks on both lines. Drag D along the transversal and CD tilts. The three angles at P never move. Only d changes.

Watch out: The pairs still exist when CD is tilted — d still has a corresponding, an alternate and a co-interior partner. What disappears is the rule that links them. That only holds while the lines are parallel.

Name the pairs

What makes each pair the pair it is?

Each pair is defined by where its two angles sit. Picture the board: two lines, a transversal, one angle at each crossing. Alternate angles come in two kinds — interior alternate (both between the two lines) and exterior alternate (both outside them). For each pair, tick every property it has, then check.

Corresponding angles
Interior alternate angles
Exterior alternate angles
Co-interior angles

Why it works

?

Reason it through

Why are interior alternate angles in parallel lines equal — and why do co-interior angles add to 180°?

Link 1 of 4

First link · your turn

Start with angle d at the bottom crossing, between the lines. Which angle at the top crossing is in exactly the same position?

2
Locked — reveal the link above first
3
Locked — reveal the link above first
4
Locked — reveal the link above first

Your turn to chase

Crossing to crossing, one reason at a time

AB and CD are parallel. A transversal crosses AB at P and CD at Q. At P, the angle below AB on the right of the transversal is 67°. At Q, find a (above CD, right), b (below CD, right) and c (above CD, left). Give a reason for every angle.

  1. Known: at P, the angle below AB on the right is 67°.It is between the lines, on the right of the transversal. So is a.
  2. missing step
Which line is step 2?

What do you think?

No arrows on the lines

A transversal crosses two lines. The lines look nearly parallel, but neither has arrow marks. At the top crossing, the angle below the top line on the left of the transversal is 104°. At the bottom crossing, the angle above the bottom line on the left is 72°.

Which is closest to what you think right now?
How sure are you?

Spot the slip

Parallel lines hiding in a trapezium

ABCD is a trapezium with AB parallel to DC. Angle DAB = 70° and angle ABC = 115°. Find angle ADC and angle BCD, giving reasons.

A student's answer — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Three angle facts that come free with parallel lines — and stop working the moment the lines aren't parallel.

What you need to know

  • A transversal is a line that crosses other lines at different points. It can cross any lines, parallel or not.
  • Parallel lines are marked with matching arrows. A second, different pair of parallel lines gets double arrows.
  • Corresponding angles: same side of the transversal, same position at each crossing. Equal when the lines are parallel.
  • Alternate angles: opposite sides of the transversal, both between the lines (interior) or both outside them (exterior). Equal when the lines are parallel.
  • Co-interior angles: same side of the transversal, both between the lines. Add to 180° when the lines are parallel.
  • Every angle you find needs its reason with its condition, such as 'corresponding angles in parallel lines are equal'.

The big picture

A transversal is a line that crosses other lines at different points. Where it crosses two lines, the angles pair up: corresponding angles (same side, same position at each crossing), alternate angles (opposite sides, both between the lines or both outside them) and co-interior angles (same side, both between the lines). If the lines are parallel, corresponding angles are equal, alternate angles are equal and co-interior angles add to 180°. If they're not parallel, none of that holds. Use the facts one after another to find unknown angles, and give each angle its fact and its condition.

Key points

1The parallel-line facts only work when the lines are parallel.
2If co-interior angles don't add to 180°, the lines are not parallel.
3'Z angles' is a weak name: exterior alternate angles don't make a Z. Say 'alternate angles'.
4Alternate angles are equal because a corresponding angle and a vertically opposite angle link them; co-interior angles then add to 180° because of angles on a straight line.
5You can find several unknown angles in a row, moving from crossing to crossing. More than one route can work.
6A side of a shape can be a transversal: in a trapezium, the co-interior angles between the parallel sides add to 180°.

Worked example

Problem

Two parallel lines are crossed by a transversal. A pair of co-interior angles are x and 2x. Find both angles.

⚠ Watch out

Treating co-interior angles as equal. Co-interior angles in parallel lines add to 180°. If you ever get two co-interior angles the same size, check whether they are both 90°.

🧠

Memory hook

Parallel first, then the fact. Corresponding and alternate: equal. Co-interior: the odd one out — adds to 180°.

✓

Check yourself

Two parallel lines are crossed by a transversal. One angle between the lines is 58°. Find its corresponding angle, its alternate angle and its co-interior angle, with a reason for each. (Answers: 58°, 58°, 122°.)

Flashcards

(12)
What is a transversal?
A line that crosses other lines at different points. The lines it crosses don't have to be parallel.
How do you mark lines as parallel on a diagram?
With matching arrow marks. A second pair of parallel lines gets double arrows so you can tell the pairs apart.
Where are corresponding angles?
On the same side of the transversal, in the same position at each crossing. In parallel lines they are equal.
Interior vs exterior alternate angles?
Both are on opposite sides of the transversal. Interior: both between the lines. Exterior: both outside them. In parallel lines, both kinds are equal.
Why is 'Z angles' not a great name?
Exterior alternate angles don't make a Z, but they are still alternate and still equal in parallel lines.
What do co-interior angles in parallel lines add up to?
180°. They're the one pair that isn't equal.
Do the angle facts work if the lines aren't parallel?
No. The pairs still exist, but they're only equal (or add to 180°) when the lines are parallel.
Co-interior angles of 95° and 88°. Parallel lines?
No. 95° + 88° = 183°, not 180°, so the lines are not parallel.
Why do co-interior angles in parallel lines add to 180°?
One of them sits on a straight line next to the alternate angle to the other. Alternate angles are equal, and angles on a straight line add to 180°.
Write a full reason for two equal corresponding angles.
'Corresponding angles in parallel lines are equal.' The words 'in parallel lines' are part of the reason.
Is there only one way to find an unknown angle in parallel lines?
No. You move from angle to angle, and different routes can work — every correct route, with correct reasons, reaches the same answer.
Why can a trapezium use parallel-line facts?
It has a pair of parallel sides, and each of the other sides is a transversal. So its co-interior angles between the parallel sides add to 180°.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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