Angles in parallel lines: alternate, corresponding and co-interior

TutorAura guide · Updated 9 November 2026

When a straight line (a transversal) crosses two parallel lines, it makes eight angles, and they come in matching pairs. Knowing which pairs are equal and which add up to 180° lets you find any missing angle from just one. This guide covers the three parallel-line rules, the other basic angle facts, and how to give the reasons examiners want.

Key takeaways

Alternate angles are equal (a Z shape).
Corresponding angles are equal (an F shape).
Co-interior angles add up to 180° (a C or U shape).
Write the proper name as your reason, not 'Z angles'.

The three parallel-line rules

Pair	Rule	Shape to look for
Alternate angles	Equal	Z (on opposite sides of the transversal, between the lines)
Corresponding angles	Equal	F (in the same position at each crossing)
Co-interior (allied) angles	Add up to 180°	C or U (on the same side, between the lines)

The shapes help you spot the pairs, but in an exam you must write the full name: "alternate angles are equal".

The basic angle facts

You'll combine the parallel-line rules with these:

Angles on a straight line add up to 180°.
Angles around a point add up to 360°.
Vertically opposite angles are equal.
Angles in a triangle add up to 180°.
Angles in a quadrilateral add up to 360°.

Example 1: one angle gives all eight

A transversal crosses two parallel lines, and one angle is 65°.

The angle vertically opposite it is 65°.
The angles next to it on the straight line are $180° - 65° = 115°$ .
At the other crossing, the corresponding angles are 65° and 115° in the matching positions.

Every one of the eight angles is either 65° or 115°, and any acute and obtuse pair adds up to 180°.

Example 2: with algebra

Co-interior: two co-interior angles are $3x + 20$ and $2x + 10$ .

3x + 20 + 2x + 10 = 180 \quad\Rightarrow\quad 5x = 150 \quad\Rightarrow\quad x = 30

The angles are 110° and 70°.

Alternate: two alternate angles are $4x - 10$ and $2x + 30$ . They're equal, so $4x - 10 = 2x + 30$ , giving $x = 20$ and an angle of 70°.

Example 3: a multi-step problem

Two parallel lines are crossed by two lines that meet to form a triangle between them. One base angle is 50° (alternate to a marked 50° angle), and the other is 180° − 120° = 60° (co-interior with a marked 120° angle). The third angle of the triangle is $180° - 50° - 60° = 70°$ .

The method: find what you can, mark each angle on the diagram as you go, and give a reason for every step.

Proving lines are parallel

The rules work both ways. If a pair of alternate angles is equal, or a pair of corresponding angles is equal, or co-interior angles add up to 180°, then the lines are parallel. For example, if co-interior angles are 105° and 75°, the lines are parallel because $105° + 75° = 180°$ .

Common mistakes

Using the rules when the lines aren't parallel (look for the arrows that mark parallel lines).
Writing "Z angles" or "F angles" as reasons: you may lose the mark.
Assuming angles that look equal are equal: diagrams are usually not drawn accurately.
Mixing up co-interior (add to 180°) with alternate (equal).

Practice questions

An angle is 72°. What is the alternate angle?
Co-interior angles are $x$ and $3x$ . Find $x$ .
Corresponding angles are $5y - 15$ and $3y + 25$ . Find $y$ and the angle.

Answers: 1. 72°. 2. $4x = 180$ , so $x = 45°$ . 3. $5y - 15 = 3y + 25$ , so $y = 20$ and the angle is 85°.

Next, see angles in polygons. Parallel-line rules are also how you find back bearings.

Frequently asked questions

What are alternate angles?

Alternate angles are on opposite sides of the transversal, between the two parallel lines. They're equal, and form a Z shape.

Do co-interior angles add up to 180?

Yes. Co-interior (allied) angles are on the same side of the transversal, between the parallel lines, and add up to 180°.

What are corresponding angles?

Corresponding angles are in the same position at each place the transversal crosses a parallel line. They're equal, and form an F shape.

Sources

GCSE mathematics: subject content and assessment objectives. Department for Education.
Cambridge IGCSE Mathematics 0580 syllabus. Cambridge International.

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