Parallel lines and transversals

A line crossing two parallel lines makes eight angles, yet only two sizes appear among them, and those two always add to 180°.

Eight angles from one crossing line

Railway tracks, the lines on writing paper and the edges of a ruler are parallel: they run side by side and never meet. A line that crosses two other lines is called a transversal. Where it cuts each line it makes four angles, so there are eight in total. OpenStax numbers them 1 to 4 at the top crossing and 5 to 8 at the bottom one, and calls angles 3, 4, 5 and 6 interior because they lie between the two lines. Angles 1, 2, 7 and 8 lie outside and are exterior.

The pair names

When the two lines are parallel, the transversal meets both of them at the same angle, so the picture at the bottom crossing is a copy of the picture at the top. That one fact gives every rule below. Corresponding angles are in the same position at the two crossings, such as 1 and 5 or 4 and 8, and they are equal. Alternate interior angles lie between the lines on opposite sides of the transversal, 3 and 6 or 4 and 5, and they are equal. Alternate exterior angles lie outside the lines on opposite sides, 2 and 7 or 1 and 8, and they are equal too.

One more pair is useful. Same-side interior angles, such as 3 and 5, lie between the lines on the same side of the transversal. They are not equal; they add to 180°. You can see why: angle 5 equals angle 1 (corresponding), and angles 1 and 3 form a linear pair.

Working out all eight from one

Because only two sizes appear, knowing one angle tells you all eight. Say angle 2 is 62°. Its vertical partner, angle 3, is 62°, and the corresponding angles 6 and 7 are 62° as well. Every other angle is the supplement, 180 − 62 = 118°. In practice you rarely need the pair names at all: decide whether an angle looks acute or obtuse, and it is either the given angle or 180° minus it.

The rules also run backwards, which gives a practical test for parallel lines. If a transversal makes equal corresponding angles with two lines, the lines are parallel. If the angles differ by even one degree, the lines will meet somewhere, as the tilt slider in the lab shows.

Letter shapes that help

A common memory aid uses letter shapes hidden in the figure. Trace alternate interior angles and the two parallel lines with the transversal between them make a Z (or a backwards Z): the equal angles sit in the two inner corners of the Z. Corresponding angles sit under the arms of an F, and same-side interior angles sit inside a C or U. The letters are only a memory aid; the reason is still that the transversal meets both parallel lines at the same angle.

A made-up street map shows the idea at work. Two parallel avenues are crossed by a diagonal road at 70°. A driver turning from the road onto either avenue makes the same sharp turn of 70° at one corner and the same wide turn of 110° at the other, because the two crossings are copies of each other.