Angles on Parallel Lines

A line crossing two parallel lines

Imagine two straight, parallel lines crossed by a third line, called a transversal. The crossings create pairs of angles at different points. Because the parallel lines run in the same direction, the transversal meets each at the same angle. This lets you transfer an angle from one crossing to the other.

Corresponding angles: matching positions

Corresponding angles occupy the same relative position at the two crossings. In the diagram, $a$ and $e$ are both above their parallel line and to the left of the transversal, so $a=e$. Likewise, $b=f$, $c=g$ and $d=h$. These equalities depend on the two crossed lines being parallel. You may recognise an F-shape, but matching positions works even when the diagram is turned around.

Alternate angles: opposite sides

Alternate angles lie on opposite sides of the transversal. An alternate interior pair lies between the parallel lines: $d=f$ and $$ in the diagram. Alternate exterior pairs lie them: and . A Z-shape can help you spot a pair, but check the positions rather than relying on the letter’s orientation.

These facts connect with angle facts you already know. For example, $d=h$ because they are corresponding angles; $h=f$ because they are vertically opposite at the lower crossing. Therefore $d=f$, showing why that alternate interior pair is equal.

Finding an unknown angle

Suppose $d=63^\circ$. The alternate interior angle $f$ is also $63^\circ$. At the lower crossing, $e$ and sit together on a straight line, so . Each step uses a different reason: first on , then angles on a straight line. If the crossed lines were not parallel, you could not assume .