Angles and parallel lines
When a straight line (a transversal) crosses two parallel lines, it makes eight angles but only two different sizes. Three named rules tell you which angles are equal and which add up to 180°.
Part 1 of 3: Learn it
In short
- 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).
Where this is in your specification
Spec points: DfE G1 and G3
| Board | Topic: Angles, polygons and parallel lines |
|---|---|
| DfE content | G1, G3, G4, G6 |
| AQA 8300 | Geometry and measures |
| Edexcel 1MA1 | Geometry and measures |
| OCR J560 | 8 Basic geometry |
| Eduqas C300 | HG1, HG3, HG4, HG6 |
| Cambridge IGCSE 0580 | E4.1, E4.6 |
| National 5 C847 75 | Geometric skills: angles in shapes |
The three rules
| Name | Shape to look for | Rule |
|---|---|---|
| Alternate | Z | equal |
| Corresponding | F | equal |
| Co-interior (allied) | C or U | add up to 180° |
Arrows on two lines in a diagram mean they are parallel. The rules only work for parallel lines.
With the basic facts
- Vertically opposite angles, where two lines cross, are equal.
- The angles on one side of a straight line total 180°.
- So once you know one angle where the transversal crosses a parallel line, you can find all eight.
Angles with algebra
If two angles are alternate or corresponding, set their expressions equal. If they are co-interior, add them and set the total to 180°. Solve, then put x back in to find the angle the question asks for.
An angle of 74° is co-interior with angle p. Find p.
Show the answer
180 − 74 = 106°.
Part 2 of 3: See it worked
Worked examples
Example 1
Two angles are alternate angles between parallel lines. One is 3x + 10 and the other is 2x + 45. Find the size of the angles.
- They are alternate, so the expressions match: 3x + 10 = 2x + 45
- x = 35
- 3 × 35 + 10 = 115
Answer: Both angles are 115°.
Example 2
Two co-interior angles are 2y + 30 and y + 15. Find both angles.
- They are co-interior, so they total 180°, giving 3y + 45 = 180
- 3y = 135, so y = 45
- 2 × 45 + 30 = 120 and 45 + 15 = 60
Answer: 120° and 60°.
Common mistakes
- Setting co-interior angles equal instead of adding them to 180°.
- Using the parallel line rules on lines that are not marked parallel.
- Writing "Z angles" as a reason.
- Giving x when the question asks for the angle.
Which pair of angles makes an F shape, and what is the rule?
Show the answer
Corresponding angles, which are equal.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
An angle of 74° is co-interior with angle p. Find p.
180 − 74 = 106°.
Which pair of angles makes an F shape, and what is the rule?
Corresponding angles, which are equal.
Two co-interior angles are x and 3x. Find x.
4x = 180, so x = 45°.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Diese Lernzettel sind auf Englisch, weil sie britischen Prüfungslehrplänen folgen.
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