Angles in Parallel Lines: Edexcel GCSE Foundation maths revision notes

Parallel lines and a crossing line

Parallel lines are always the same distance apart, so they never meet. On a diagram, matching arrows on two lines show that they are parallel.

When a straight line crosses two parallel lines, it makes four angles at each crossing, and the angles at one crossing match the angles at the other.

Two facts work at any crossing: vertically opposite angles are equal, and angles on a straight line add up to 180^\circ. Parallel lines give you three more facts, which link the angles at one crossing to the angles at the other.

In the exam you are usually asked to find a missing angle and give a reason, for 1 or 2 marks. Harder questions, for 4 or 5 marks, put parallel lines with a triangle or a quadrilateral and say ‘Give a reason for each stage of your working’. The diagrams are not drawn accurately, so work the angles out and never measure them.

Vertically opposite angles are equal

The two marked angles are opposite each other at the same crossing. They stay equal as the line turns.

Angles on a straight line add up to 180°

These two angles meet at the same point and together make a straight angle. As one gets larger, the other gets smaller by the same amount.

Corresponding angles are equal

Corresponding angles are in matching positions at the two crossings. Here, both angles are above their line and on the right of the crossing line.

Corresponding angles are equal. They sit inside an F shape, which may be turned round or back to front.

Worked example 1: Corresponding angles

The two lines with arrows are parallel.

Work out the value of x.

Both angles are above their parallel line and on the left of the crossing line, so they are corresponding angles.
Your turn. Corresponding angles are equal. What is the value of x?

Your answer

x=62
Answer: x=62, because corresponding angles are equal.

Co-interior angles add up to 180°

Co-interior angles are both between the parallel lines, on the same side of the crossing line. They sit inside a C shape. Some books call them allied angles.

Co-interior angles add up to 180^\circ. They are not equal: one is acute and the other is obtuse, unless both are right angles.

If co-interior angles are written as expressions, add them and make the total equal to 180. Then solve the equation.

Worked example 2: Co-interior angles

The two lines with arrows are parallel.

Work out the value of x.

Both angles are between the parallel lines and on the left of the crossing line, so they are co-interior angles.
Co-interior angles add up to 180^\circ, so x+121=180
Your turn. Work out 180-121

Your answer

x=180-121=59
Check: 59+121=180
Answer: x=59, because co-interior angles add up to 180^\circ.

Alternate angles are equal

Alternate angles are both between the parallel lines, on opposite sides of the crossing line. They sit inside a Z shape, which may be back to front.

Alternate angles are equal.

The F, Z and C shapes only help you spot the angles. In the exam, write the reason in full: ‘alternate angles are equal’, ‘corresponding angles are equal’ or ‘co-interior angles add up to 180^\circ’ (‘allied angles’ is accepted too). The mark scheme looks for those key words, so ‘Z angles’, or just ‘parallel lines’, does not get the mark for a reason.

Worked example 3: Two angle facts

The two lines with arrows are parallel.

Work out the value of x. Give a reason for each stage of your working.

Start from the 65^\circ angle. Its alternate angle is at the lower crossing, above the line and on the left of the crossing line.
That angle is 65^\circ, because alternate angles are equal.
That 65^\circ angle and x meet at the same point, on the crossing line, which is straight.
Your turn. Angles on a straight line add up to 180^\circ. What is the value of x?

Your answer

x=180-65=115
Answer: x=115. Reasons: alternate angles are equal, and angles on a straight line add up to 180^\circ.

Worked example 4: Angles written as expressions

The two lines with arrows are parallel.

Work out the value of x and the size of each marked angle.

The two angles are between the parallel lines, on opposite sides of the crossing line. They are alternate angles, so they are equal.
Write an equation: 5x-12=3x+20
Subtract 3x from both sides, then add 12 to both sides: 2x=32
Your turn. What is the value of x?

Your answer

x=32\div2=16
Each angle is 5\times16-12=68^\circ. Check: 3\times16+20=68
Answer: x=16, and each angle is 68^\circ

Parallel lines with triangles

In longer questions the parallel lines are part of a triangle or a quadrilateral. The side of a triangle can be the line that crosses both parallel lines.

Work out one angle at a time. Write each angle on the diagram, or name it with three letters: angle ABC is the angle at B, between the lines BA and BC.

Give the reason for each angle as you find it, such as ‘angles in a triangle add up to 180^\circ’. Only use what you are told: do not treat a triangle as isosceles unless the question says so or two sides are marked equal.

Worked example 5: Parallel lines and a triangle

AB and CD are parallel lines. E is a point on AB. F and G are points on CD.

Angle AEF =105^\circ and angle EGF =35^\circ

Work out the size of angle FEG, marked x. Give a reason for each stage of your working.

The line EG crosses both parallel lines. Angle BEG and angle EGF are on opposite sides of it, between the parallel lines, so they are alternate angles.
Angle BEG =35^\circ, because alternate angles are equal.
AEB is a straight line, so angles AEF, FEG and BEG add up to 180^\circ.
Your turn. Work out 180-105-35

Your answer

x=180-105-35=40
Check: angle EFG =105^\circ (alternate to angle AEF), and in triangle EFG, 105+35+40=180
Answer: x=40. Reasons: alternate angles are equal, and angles on a straight line add up to 180^\circ.

Showing that lines are parallel

The three facts also work backwards. If two corresponding angles are equal, or two alternate angles are equal, or two co-interior angles add up to 180^\circ, then the lines are parallel.

When a question says ‘Show that AB is parallel to DE’, do not start by assuming that the lines are parallel. Work out the angles you need from the other facts. Then say which pair of angles proves it, and why.

If the pair does not fit the fact, the lines are not parallel. For example, co-interior angles of 115^\circ and 68^\circ add up to 183^\circ, not 180^\circ, so those lines are not parallel.

Worked example 6: Showing that lines are parallel

FAB, ACE and BCD are straight lines.

Angle FAC =116^\circ, angle CDE =48^\circ and angle DCE =68^\circ

Show that AB is parallel to DE. Give a reason for each stage of your working.

Start with triangle CDE, where two angles are known.
Your turn. Angles in a triangle add up to 180^\circ. Work out angle CED.

Your answer

Angle CED =180-48-68=64^\circ
FAB is a straight line, so angle BAC =180-116=64^\circ, because angles on a straight line add up to 180^\circ.
The line AE crosses AB and DE. Angle BAC and angle CED are on opposite sides of it, between AB and DE, so they are alternate angles.
Angle BAC =64^\circ and angle CED =64^\circ, so the alternate angles are equal.
Answer: the alternate angles BAC and CED are equal, so AB is parallel to DE.

Common mistakes

Giving ‘parallel lines’ as the reason. Name the angle fact in full, such as ‘alternate angles are equal’.
Writing half a reason. ‘Opposite’ or ‘triangle is 180’ is not enough: write ‘vertically opposite angles are equal’ or ‘angles in a triangle add up to 180^\circ’.
Making co-interior angles equal. Co-interior angles add up to 180^\circ, so take the angle you know away from 180.
Assuming facts you are not given. Lines are only parallel, and a triangle is only isosceles, if the question or the marks on the diagram say so.
Using a straight line for angles at different points. Angles on a straight line add up to 180^\circ only when they meet at the same point.
Not saying which angle you mean. Write each angle on the diagram, or name it with three letters, such as angle ABC.