Angles, Lines and Parallel Lines

What you will learn

• Angles on a straight line add to 180°.
• Angles around a point add to 360°.
• Vertically opposite angles are equal.
• Corresponding and alternate angles are equal when lines are parallel.
• Co-interior angles add to 180° when lines are parallel.
• How to write short reasons for angle answers, including when the diagram is not drawn accurately.
• How to form and solve equations when angle expressions are equal or add to 180°.

Why this matters

Angle facts are used in polygons, bearings, constructions, circle theorems, trigonometry and proof. A tidy angle reason can turn a guessed-looking answer into a full-mark method.

Prior knowledge

You should already be comfortable with:

• acute, right, obtuse and reflex angles,
• adding and subtracting from 180 and 360,
• solving simple equations such as 2x + 30 = 180,
• reading labels on a diagram carefully.

Clear explanation

Three facts you use constantly

• Straight line: angles on a straight line add to 180°.
• Around a point: angles around a point add to 360°.
• Vertically opposite: opposite angles made by two crossing straight lines are equal.

Parallel-line angle facts

When a line crosses two parallel lines, it creates repeatable patterns. Many pupils remember these by shape:

• Corresponding angles are equal: look for an F shape and match the same relative position at each crossing.
• Alternate angles are equal: look for a Z shape between the parallel lines.
• Co-interior angles add to 180°: look for a C shape inside the parallel lines.

In exams, the reason matters. Write something like alternate angles are equal because the lines are parallel. If the lines are not marked parallel, do not use the F, Z or C-shape rules unless the question tells you they are parallel.

Treat the diagram as a clue, not a measuring tool. If the question says the drawing is not to scale, use the angle labels and facts only.

Worked examples

Quick checks

Choose an answer, then check your thinking.

Practice questions

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For angle questions, marks usually come from selecting the correct angle fact, doing the subtraction or equation accurately, and writing a clear reason such as angles on a straight line, vertically opposite angles, corresponding angles, alternate angles or co-interior angles. When angle expressions include x, form the equation from the angle fact before solving. Do not measure a not-to-scale diagram.

Common mistakes

• Measuring the printed diagram: GCSE diagrams are often not drawn accurately unless the question says they are.
• Using a parallel-line rule without parallel lines: corresponding, alternate and co-interior facts need parallel lines.
• Mixing up equal and sum-to-180 facts: corresponding and alternate are equal; co-interior add to 180°.
• Forgetting reasons: write the angle fact, not just the answer.

Extension challenge

A pair of co-interior angles are labelled 2x + 15 and 5x − 10. Find both angles.

Exam-board guidance

Angle facts and parallel-line reasoning are assessed across GCSE Maths boards. The shared habit is to pair each number with a reason, especially when a question asks you to explain or prove, to say when a rule depends on parallel lines, and to trust written labels over measuring not-to-scale drawings.

Next lesson

Next, use angle facts inside shapes in Polygons and Interior/Exterior Angles.