IGCSE Maths / Working with shape and space

Angles in parallel lines revision notes

Edexcel International GCSE Maths (4MA1), chapter 11: Working with shape and space, specification point 4.1b. Free lesson.

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When a straight line crosses two parallel lines, three angle relationships help you find missing angles. Identifying the pair and naming its property in full can earn both the answer and the reason marks.

Corresponding angles are in the same corner

A transversal is a straight line that crosses two other lines. When it crosses parallel lines, corresponding angles sit in the same corner at each crossing. Check that they are both in the same position before treating them as equal. Parallel lines are marked with matching arrows. Corresponding angles between parallel lines are equal.

Alternate angles are on opposite sides

Alternate angles lie between the parallel lines and on opposite sides of the crossing line. They form a Z shape, which can help you spot them, but the shape is not the name of the rule. Alternate angles between parallel lines are equal. At T the given angle is 47°. The angle x at U is on the opposite side of the crossing line and inside the parallel pair, so x = 47°. The other angle at U on the straight line is 180° − 47° = 133°. It lies inside the pair on the same side as the 47° angle.

Co-interior angles are on the same side

Co-interior angles lie between the parallel lines and on the same side of the crossing line. They are the odd one out: they add to 180° rather than being equal. Allied angles is another name for this property. At U, the given angle is 117°. The angle x at T is inside the parallel pair and on the same side of the crossing line. Therefore x = 180° − 117° = 63°, because co-interior angles between parallel lines add up to 180°. The angle b on the other side of T is 117°, since it completes a straight line with x.

Choose the pair before calculating

First check whether the angles are inside or outside the parallel lines, then check which side of the crossing line they are on. This tells you whether they are equal or add to 180°. Same corner at each crossing: corresponding, so equal. Inside the pair, opposite sides: alternate, so equal. Inside the pair, same side: co-interior, so add to 180°. For the quick check, the angle is 82° and the wanted angle is inside the pair on the opposite side of the crossing line. They are alternate angles, so the answer is 82°.

Give the full reason

A letter shape such as F, Z or C helps you recognise an angle pair, but it is not a sufficient written reason. Write the property in full to earn the reason mark. Use statements such as “alternate angles between parallel lines are equal”. For co-interior angles, state that they add up to 180°.