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Angles in parallel lines

Part of MathsGeometry and measure

Key points about angles in parallel lines

Bullet points represented by lightbulbs

  • are two or more straight lines that remain the same distance apart and never intersect.

  • A is a line which crosses two or more parallel lines. The point where it crosses is called a .

  • When a transversal intersects a pair of parallel lines, various types of angles are formed within the parallel lines: , and .

Make sure you are confident with solving linear equations before working with angles written as algebraic expressions.

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Check your understanding

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What are alternate angles?

An illustration of the parallel lines with a diagonal line through the middle of both. Around the point of each intersection, acute angles are orange and obtuse angles are in blue.

When a transversal intersects a pair of parallel lines, the angles at both points of intersection are related.

Along a specific transversal, all of the are the same size.

All of the are the same size.

Pairs of angles can be given special names.

Alternate angles are on opposite sides of the transversal within the parallel lines.

  • Alternate angles are always equal in size.
  • When looking for alternate angles, it can useful to look for a Z-shape.
  • The Z-shape can be backwards, sideways or upside down.
An illustration of the parallel lines with a diagonal line through the middle of both. Around the point of each intersection, acute angles are orange and obtuse angles are in blue.
An example of the 'Z' shaped formed with alternate angles.
An example of the 'Z' shaped formed backwards with alternate angles.
An example of the 'Z' shaped formed sideways with alternate angles.

Follow the worked example below

GCSE exam-style questions

A pen and a piece of paper with question marks on it.
  1. Work out the size of angle π‘š.
A pair of parallel lines going right to left with a slightly diagonal line down the middle creating angles m, 71Β° and 109Β°.

  1. Which pairs of angles are alternate?
A pair of parallel lines going upwards with a a line going diagonal left to right creating angles a and c above the line and b and d below.

  1. Set up and solve an equation to find π‘₯.
A pair of parallel lines going from left to right, with a diagonal line intersecting. On the left hand side of the line at the bottom the angle is 115Β° and in the diagonal angle opposite (in orange) 3π‘₯ – 8.

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What are corresponding angles?

An illustration showing how the angles occur at the same position within each intersection between a diagonal line going through a pair of horizontal parallel lines. The 'f' shape is highlighted in different colours across the four different illustrations.

Corresponding angles occur at the same position within each intersection.

  • Corresponding angles are always equal in size.

  • When looking for corresponding angles, it can be helpful to look for an F- shape.

  • The F-shape can be backwards, sideways or upside down.

An illustration showing how the angles occur at the same position within each intersection between a diagonal line going through a pair of horizontal parallel lines. The 'f' shape is highlighted in different colours across the four different illustrations.

Follow the worked example below

GCSE exam-style questions

A pen and a piece of paper with question marks on it.
  1. 𝑃𝑄 is parallel to 𝑅𝑆.

Work out the size of angle 𝑦.

A geometric diagram featuring two triangles, PQT and QRS, sharing a common side. Triangle PQT is larger and contains triangle QRS within it. Both triangles have a right angle at point Q. Angle SQR is marked as 43Β° and angle PQT is labeled as "y" in orange. The vertices are labeled P, Q, R, S, and T.

  1. Which pairs of angles are corresponding?
A geometric diagram showing two parallel vertical lines intersected by a single diagonal line. The intersections form four angles labelled a, b, c, and d. Angles a and b are on the left side of the diagonal, while angles c and d are on the right side.

  1. Set up and solve an equation to find 𝑦.
A geometric diagram showing two parallel lines intersected by a transversal line, forming two alternate interior angles. The angle on the left is labelled "3y - 31" and the angle on the right is labelled "y + 23.

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What are co-interior angles?

An illustration showing the four different ways co-interior angles can be found on a horizontal straight line with a pair of parallel lines crossing it slight diagonally vertically. Each co-interior angle is shown in half blue and half orange.
These images all show pairs of co-interior angles.

Co-interior angles (or allied angles) occur on the same side of the transversal, between the two parallel lines.

  • Co-interior angles add up to 180Β°.
  • When looking for corresponding angles, it can be helpful to look for a C-shape.
  • The C-shape can be backwards, sideways or upside down.
An illustration showing the four different ways co-interior angles can be found on a horizontal straight line with a pair of parallel lines crossing it slight diagonally vertically. Each co-interior angle is shown in half blue and half orange.
These images all show pairs of co-interior angles.

Follow the worked example below

GCSE exam-style questions

A pen and a piece of paper with question marks on it.
  1. Trapezium 𝐴𝐡𝐢𝐸 is made from parallelogram 𝐴𝐡𝐢𝐷 and isosceles triangle 𝐴𝐷𝐸.

𝐴𝐸 = 𝐷𝐸

Work out the size of angle 𝐴𝐸𝐷.

A diagram showing parallelogram ABCD and isosceles triangle AED to form a trapezium. Angle ABC is labelled as 125Β° and students are asked to find angle AED.

  1. Which pairs of angles are co-interior?
A geometric diagram showing two parallel vertical lines intersected by a single diagonal line. The intersections form four angles labelled a, b, c, and d. Angles a and b are on the left side of the diagonal, while angles c and d are on the right side.

  1. Set up and solve an equation to evaluate 𝑧.
A pair of horizontal parallel lines with a vertical line going slightly diagonally with the equations show on the left hand side of each intersection.

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Quiz - Angles in parallel lines

Practise what you've learned about angles in parallel lines with this quiz.

Now you've revised angles in parallel lines, why not look at bearings?

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