One agreement, three results

A line cuts two parallel lines. How are the eight angles related?

The rule

Three new kinds of pair appear. Two are equal. One totals 180°.

A transversal is a line that cuts two other lines. It cuts them at two different points.

Corresponding angles are in the same corner at each crossing. For example, both are top right.

Axiom. If two lines are parallel, corresponding angles are equal. We accept this without proof.

For parallel lines, two more facts follow from the axiom.

• Alternate angles are between the lines, on opposite sides of the transversal. They are equal. Reason: one alternate angle is vertically opposite another angle. That angle corresponds to the second alternate angle. So the two are equal.
• Co-interior angles are between the lines. They are on the same side of the transversal. They total 180°. Reason: one co-interior angle makes a linear pair with a second angle. Those two total 180°. The second angle corresponds to the other co-interior angle. So the second angle equals the other co-interior angle. So the two co-interior angles total 180°.