Alternate angles

Another useful pair: alternate angles. In the figure, angle d is the alternate angle of f, and c of e. They lie between the two lines, on opposite sides of the transversal. (Trace a Z: the two angles in its corners are alternate.)

To find the alternate angle of f in two steps: first its corresponding angle, b; then the angle vertically opposite b, which is d.

Those two steps also prove that alternate angles are equal when the lines are parallel. Say f = 120°. Its corresponding angle b is also 120° (lines parallel). d is vertically opposite b, so d = 120°.

Now without any number: f = b (corresponding angles, lines parallel), and b = d (vertically opposite). So f = d, always.

Alternate angles formed by a transversal across two parallel lines are always equal. And look how we got there: a new fact, built from two facts we had already proved. That's how geometry grows.