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Parallel Lines

When a straight line cuts across two parallel lines it creates a family of matching angles. Learn the three relationships and you can find every angle in the diagram from just one.

MYP 4Standard MathsGeometryCriteria A · C · D~10 min read

Written by Fayek Jawad Haider · MYP student

The transversal setup

Parallel lines are lines that stay the same distance apart and never meet, shown by small matching arrows. A transversal is a straight line that crosses them.

Transversal
A line that cuts across two or more other lines. Where it crosses two parallel lines it makes eight angles, and they come in a small number of equal or supplementary pairs.

Every one of the eight angles is either equal to a given angle or makes \(180^\circ\) with it. The three named relationships below tell you which is which.

Corresponding angles

Corresponding angles sit in the same position at each crossing, for instance both above the parallel line and both on the left of the transversal. They are equal. The shape they trace out looks like the letter F.

\[ \text{corresponding angles are equal} \]
Worked example

A transversal crosses two parallel lines. At the upper crossing an angle of \(70^\circ\) is marked. Find the corresponding angle at the lower crossing.

1
The two angles are in the same position at each crossing, so they are corresponding angles.
2
Corresponding angles between parallel lines are equal, so the angle equals \(70^\circ\).
The corresponding angle is \(70^\circ\)

Hunt for the letter shapes

Corresponding angles make an F, alternate angles make a Z, and co-interior angles make a C or U. The letters can be back to front or upside down and still count.

Alternate angles

Alternate angles sit on opposite sides of the transversal and between the two parallel lines. They are equal, and they trace out the letter Z.

\[ \text{alternate angles are equal} \]
Worked example

Between two parallel lines, one angle made by the transversal is \(130^\circ\). Find its alternate angle.

1
The alternate angle is on the opposite side of the transversal, between the parallel lines, forming a Z.
2
Alternate angles between parallel lines are equal.
The alternate angle is \(130^\circ\)

Co-interior angles

Co-interior angles (sometimes called allied angles) sit on the same side of the transversal and between the two parallel lines. They are not equal; instead they add up to \(180^\circ\). They trace out the letter C or U.

\[ \text{co-interior angles} = 180^\circ \]
Worked example

Two parallel lines are cut by a transversal. One co-interior angle is \(68^\circ\). Find the other.

1
Co-interior angles lie on the same side of the transversal and add to \(180^\circ\): \(68 + x = 180\).
2
Subtract: \(x = 180 - 68\).
The other co-interior angle is \(112^\circ\)

Where this is assessed

Naming the correct relationship is Criterion A (Knowing and understanding). Writing the reason, such as "alternate angles are equal", beside each step is Criterion C (Communicating). Chained diagrams where you find several angles in turn belong to Criterion D (Applying maths in context).

Check yourself

1. A transversal crosses two parallel lines. One angle is \(115^\circ\). Find the corresponding angle at the other crossing. +

Corresponding angles between parallel lines are equal, so the answer is \(115^\circ\).

2. Two parallel lines are cut by a transversal. One co-interior angle is \(68^\circ\). Find the other co-interior angle. +

Co-interior angles add to \(180^\circ\), so the other is \(180 - 68 = \mathbf{112^\circ}\).

3. Between two parallel lines a transversal makes an angle of \(130^\circ\). What is its alternate angle? +

Alternate angles are equal, so it is \(130^\circ\).

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