Parallel and Intersecting Lines

When two straight lines cross each other at a single point, we say they intersect, and this crossing always creates exactly four angles around that point. Label them a, b, c, d going around the intersection in order. Since a and b sit on a straight line together, they must add up to a full straight angle: a + b = 180 degrees. A pair of adjacent angles that add up to 180 degrees like this is called a linear pair.

Now notice angles a and c, which sit directly across from each other through the intersection point — these are called vertically opposite angles. Here's a quick proof that they must always be equal, for any intersecting pair of lines: since a and b form a linear pair, a + b = 180. Since b and c also form a linear pair (b sits between a and c), b + c = 180 as well. Both equations equal 180 and both contain b, so a and c must be equal to each other. The same reasoning shows b and d are equal too.

For example, if angle a measures 70 degrees, its linear pair partner b must measure 180 - 70 = 110 degrees. The angle vertically opposite to a (angle c) stays equal to a, so it also measures 70 degrees, and the angle vertically opposite to b (angle d) matches b at 110 degrees. This works for any measure of a — the relationship a = c and b = d holds true no matter what the actual angle is.

What happens if all four angles formed by two intersecting lines are equal to each other? Since each pair of adjacent angles is a linear pair (adding to 180 degrees), and all four angles are equal, each one must be exactly half of 180 degrees: 90 degrees. Two lines that intersect to form four right angles like this are called perpendicular lines.

You've already been using perpendicularity without necessarily naming it — the adjacent edges of this very page meet at a right angle, which means they're perpendicular to each other. In diagrams, perpendicular lines are marked with a small square placed right at the vertex where they meet, instead of writing '90 degrees' every single time.

Perpendicularity is also the quickest way to build other right angles accurately using simple tools. A set square (the triangular ruler with a right-angle corner) can be slid along a ruler to draw a line perpendicular to a given line at any point you choose — a technique you'll use again when drawing parallel lines later in this chapter.

Not every pair of lines on a flat surface intersects. Two lines that lie in the same plane and never meet, no matter how far they are extended in either direction, are called parallel lines. The opposite edges of a straight-edged notebook, the rungs of a ladder, and the rails of a railway track are everyday examples of lines that stay parallel.

That phrase 'in the same plane' matters more than it might seem. A line drawn on your desk and a line drawn on the wall behind it might never cross either, but that alone doesn't make them parallel — parallel lines must lie flat within one single plane together, like two lines drawn on the same sheet of paper.

Because it's impossible to check 'never meet, however far extended' by actually drawing lines forever, mathematicians use a notation instead: a single arrow mark (>) placed on each of a pair of parallel lines shows that they are parallel to each other. If a diagram has a second, separate pair of parallel lines, that pair is marked with double arrows (>>) instead, keeping the two parallel pairs visually distinct.

So far we've looked at what happens at a single intersection. Now consider a line t that crosses two separate lines, l and m — this crossing line t is called a transversal. Because t intersects two lines, it creates two separate sets of four angles (one set where t meets l, another where t meets m), for a total of eight angles. Since vertically opposite angles within each set of four must be equal, there can be at most four distinct angle measures among all eight angles, however the transversal is drawn.

Of the four angles formed at each intersection, some occupy matching positions relative to the transversal and the two lines — for example, both in the upper-left position relative to where the transversal crosses each line. Angles in matching positions like this are called corresponding angles.

Here's the central fact of this chapter: if a transversal crosses two lines and a pair of corresponding angles are equal, then the two lines are parallel — and the reverse is just as true: if two lines are already known to be parallel, every pair of corresponding angles formed by any transversal crossing them will be equal. This works both ways, which makes corresponding angles the most direct test available for checking whether two lines are parallel, even without extending them.

Besides corresponding angles, a transversal crossing two lines creates two other useful angle relationships. Alternate angles lie on opposite sides of the transversal, both tucked between the two lines (one on top, one below) — and when the two lines are parallel, alternate angles are always equal. This follows directly from what you already know: an alternate angle can always be reached by first finding its corresponding angle (equal, since the lines are parallel) and then finding that angle's vertically opposite angle (also equal) — two equalities chained together, so the original and the alternate angle must match.

Suppose a transversal crosses two parallel lines and one of the angles measures 135 degrees. Its alternate angle on the parallel line, also measures 135 degrees (equal, by the chain of reasoning above). Its co-interior angle — also called an allied angle, meaning the angle on the same side of the transversal and between the two lines — is different: co-interior angles always add up to 180 degrees when the lines are parallel, so this one measures 180 - 135 = 45 degrees.

These relationships also work as tests. If a transversal crosses two lines and you measure a pair of corresponding angles as 120 degrees and 55 degrees, these are clearly not equal (120 is not 55), so the two lines are not parallel — no alternate-angle or co-interior calculation is even needed once a single pair of corresponding angles fails to match.