Year 8 · Mathematics · Space
Angles and parallel lines
Identify corresponding, alternate and co-interior angles formed when a line crosses parallel lines, and use them to find unknown angles with reasons.
Awaiting educator review. This material was drafted by our content team with AI assistance and is waiting for review by a qualified educator. Please check it suits your child before using it. Created 10 October 2026. How we review
These lessons were written around Australian Curriculum strands and matched to UK year groups by age (Year 1 is the year a child turns 6 in England). They are not mapped to the National Curriculum in England or the curricula of Scotland, Wales or Northern Ireland, so check them against your own plans.
Learning goals
Learners will
- Identify corresponding, alternate and co-interior angles formed by a transversal crossing parallel lines.
- Use these angle relationships to find unknown angles.
- Give a geometric reason for each step when finding an angle.
Success looks like
- I can point to corresponding (F), alternate (Z) and co-interior (C) angles on a diagram.
- I can find unknown angles, knowing that corresponding and alternate angles are equal and co-interior angles add to 180°.
- I can write a reason, such as “alternate angles, parallel lines”, next to each answer.
The big idea
When a line called a transversal crosses two parallel lines, it makes eight angles — four at each crossing point. Because the parallel lines run in the same direction, the two crossings look exactly the same. This gives us some special pairs of angles.
First, two facts you already know that work at any crossing: angles on a straight line add to 180°, and vertically opposite angles (the angles opposite each other in an X) are equal.
Corresponding angles are in the same position at each crossing — for example, both above the parallel line and to the right of the transversal. Trace them and you see an F shape (it may be backwards or upside down). When the lines are parallel, corresponding angles are equal.
Alternate angles are between the parallel lines but on opposite sides of the transversal. They form a Z shape (or a backwards Z). When the lines are parallel, alternate angles are equal.
Co-interior angles are between the parallel lines on the same side of the transversal. They form a C or U shape. When the lines are parallel, co-interior angles are supplementary — they add to 180°.
In geometry, an answer needs a reason. Write the relationship and the condition, for example: “∠b = 65° (corresponding angles, parallel lines)”.
The rules also work backwards. If a pair of corresponding or alternate angles are equal, or a pair of co-interior angles add to 180°, then the two lines must be parallel. This is how we test whether lines are parallel.
Worked examples
Corresponding and alternate angles
A transversal crosses two parallel lines. The angle above the top line, to the right of the transversal, is 65°. Find the corresponding angle at the bottom line, and the alternate angle to the angle below the top line on the left.
- The corresponding angle sits above the bottom line, to the right of the transversal (an F shape). Corresponding angles are equal, so it is 65°.
- The angle below the top line on the left is vertically opposite the 65° angle, so it is also 65°.
- Its alternate angle sits above the bottom line, to the right of the transversal (a Z shape). Alternate angles are equal, so it is 65° too.
Answer: Both angles are 65° (corresponding angles, parallel lines; alternate angles, parallel lines).
Co-interior angles
Two angles between parallel lines, on the same side of a transversal, are 110° and x°. Find x.
- The angles are co-interior, so they add to 180°.
- x + 110 = 180.
- x = 180 − 110 = 70.
Answer: x = 70 (co-interior angles, parallel lines)
Using an equation
A pair of alternate angles on parallel lines are (2x + 10)° and 70°. Find x.
- Alternate angles on parallel lines are equal, so 2x + 10 = 70.
- Subtract 10 from both sides: 2x = 60.
- Divide both sides by 2: x = 30.
- Check: 2 × 30 + 10 = 70. ✓
Answer: x = 30
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
Angles that are between two parallel lines and on opposite sides of the transversal (forming a Z shape) are called:
Alternate angles — Alternate angles form a Z shape and are equal when the lines are parallel.
When the lines are parallel, co-interior angles:
add to 180° — Co-interior angles (C shape) are supplementary, so they add to 180°.
One of a pair of corresponding angles on parallel lines is 48°. What is the other?
48° — Corresponding angles on parallel lines are equal, so the other angle is 48°.
Two co-interior angles on parallel lines are 75° and y°. What is y?
105 — Co-interior angles add to 180°: y = 180 − 75 = 105.
One of a pair of alternate angles on parallel lines is 117°. What is the size of the other angle, in degrees?
117° — Alternate angles on parallel lines are equal, so it is also 117°.
One of a pair of co-interior angles on parallel lines is 64°. What is the size of the other angle, in degrees?
116° — Co-interior angles add to 180°: 180 − 64 = 116°.
A pair of corresponding angles on parallel lines are (3x)° and 81°. Find x.
x = 27 — Corresponding angles are equal: 3x = 81, so x = 81 ÷ 3 = 27.
A transversal crosses two lines. A pair of co-interior angles measure 95° and 85°. Are the two lines parallel? Answer yes or no.
Yes — 95° + 85° = 180°. Co-interior angles that add to 180° show that the lines are parallel.
Watch out for
- Thinking co-interior angles are equal. Draw a C shape and measure: one angle is acute and the other obtuse, and they add to 180°.
- Using the rules when the lines are not parallel. The F, Z and C rules only work if the lines are marked parallel (or are known to be parallel).
- Giving an answer without a reason. In geometry the reason is part of the answer — encourage “F, Z, C” plus the words “parallel lines”.
Tips for parents
- Use the lines on ruled paper as ready-made parallel lines. Draw a slanted transversal and let the learner measure all eight angles with a protractor to discover the patterns.
- Have the learner highlight F, Z and C shapes in different colours on the same diagram.
- Look for parallel lines and transversals in everyday life — railway tracks and sleepers, window frames, road markings and bridges.
Go further
- Use alternate angles to explain why the angles in any triangle add to 180°: draw a line through one vertex parallel to the opposite side.
- Design a “find all the angles” puzzle with two transversals crossing a pair of parallel lines, and give it to a family member to solve.
- Investigate what happens to the angles when the two lines are not parallel — which pairs are no longer equal?
About 45 minutes
You'll need
- Ruled paper (the printed lines are parallel)
- Ruler, pencil and protractor
- Coloured pencils or highlighters
Key words
- Parallel lines
- Lines in the same plane that never meet and are always the same distance apart. They are marked with matching arrows.
- Transversal
- A line that crosses two or more other lines.
- Corresponding angles
- Angles in matching positions at each intersection, forming an F shape. They are equal when the lines are parallel.
- Alternate angles
- Angles between the parallel lines on opposite sides of the transversal, forming a Z shape. They are equal when the lines are parallel.
- Co-interior angles
- Angles between the parallel lines on the same side of the transversal, forming a C or U shape. They add to 180° when the lines are parallel.
- Supplementary angles
- Two angles that add to 180°.
Still tricky?
For parents
Want an outside view of where your child is at?
Book a free, no-obligation English and maths assessment at a Success Tutoring centre and get a clear starting point.
Book a free assessmentRelated space lessons
Build up to it (Year 7)
Next steps (Year 9)