Transversals

So far one line crossed one line. Now let one line cross two lines.

Line t crosses lines l and m. A line that crosses two (or more) lines at different points is called a transversal.

Count the angles: 4 where t crosses l, and 4 where t crosses m. Eight angles. Number them 1 to 8.

Could all eight be different sizes? Think first. At each crossing, the vertically opposite angles are equal, and we proved that works for any two crossing lines.

• At l: angle 1 = angle 3, angle 2 = angle 4.
• At m: angle 5 = angle 7, angle 6 = angle 8.

That makes four pairs of equal angles. So the eight angles can have at most four different sizes. All eight different is impossible, and so are five different sizes.

When the two lines are parallel, the number of different sizes drops to just two, as you'll see next.