Angles in Parallel Lines

Grade 4

What happens when a line crosses parallel lines

When a straight line (called a transversal) crosses two parallel lines, it creates a set of angles with reliable relationships. You need to recognise three types.

Alternate angles are equal — they sit on opposite sides of the transversal in a Z-shape. Corresponding angles are equal — they sit in the same position at each parallel line in an F-shape. Co-interior angles (also called allied angles) add up to 180° — they sit on the same side of the transversal in a C-shape.

A memory trick

Z = alternate (equal). F = corresponding (equal). C = co-interior (supplementary, add to 180°). Draw the letter shape over the diagram to spot which type you are looking at.

Worked example

EXAMPLE
Two parallel lines are cut by a transversal. One angle formed is 115°. Find the alternate angle and the co-interior angle on the same side.
Alternate angles are equal, so the alternate angle = 115°. Co-interior angles add to 180°, so the co-interior angle = 180° − 115° = 65°.

Watch out for the direction of the arrows

Arrows on lines in diagrams mark them as parallel. If there are no arrows, you cannot assume the lines are parallel — you would need to prove it first or use a different approach.
Nova
Try this one

The diagram shows a straight line XY and two lines PA and QB that are parallel to each other. A transversal meets PA at point G and QB at point H. Angle PGH = \( (9k - 5)^\circ \) and angle GHB = \( (6k + 25)^\circ \).

PAQBGH\((9k-5)°\)\((6k+25)°\)Not drawn accurately

Angle PGH and angle GHB are alternate angles.

Find the value of \( k \) and the size of angle GHB. [4]

Nova's hint:

The question tells you these are alternate angles between parallel lines.

Ask yourself: 'What is always true about alternate angles — do they add to 180° or are they equal to each other?' Use that fact to set up an equation, then solve for k and substitute back.

Try to Solve it with Nova?