Parallel Lines & Transversals

Corresponding and alternate angles are equal; co-interior angles are supplementary.

The idea

A transversal is a line that crosses two other lines, forming eight angles at its two crossings. When the two crossed lines are parallel, the eight angles take only two sizes, so one known angle determines the other seven.

The reason is that parallel lines meet the transversal at the same tilt, so the second crossing repeats the first, slid along the transversal. Three named pairs record the consequences. Corresponding angles occupy the same position at the two crossings, so they are equal. Alternate interior angles lie between the two lines on opposite sides of the transversal; each is the vertical angle of a corresponding angle, so these are equal as well. Co-interior (same-side interior) angles lie between the lines on the same side of the transversal; one lies along a straight line beside a copy of the other, so they are supplementary: they add to $180°$.

None of these relations holds if the two lines are not parallel.

Ways to work on it

• Walkthrough. Name the angle pairs and find every angle from one given measure.
• Practice. One angle and a named relationship — find the other.
• Hardest. Set up and solve an equation from an equal or supplementary pair.

Not sure where to start? Take the ten-question placement test.