Angles and parallel lines

Three pairs

Parallel lines go in the same direction and never meet; diagrams mark them with matching arrows.

A line that crosses them is called a transversal. When a line crosses two parallel lines:

• Alternate angles (a Z shape) are equal. They are between the parallel lines, on opposite sides of the crossing line.
• Corresponding angles (an F shape) are equal. They sit in the same position at each crossing.
• Co-interior angles (a C or U shape) add up to 180°. They are between the parallel lines, on the same side of the crossing line.

Here a and 70° are alternate, so a = 70°.

Often you need two steps: one parallel-line fact, then another angle fact. If an angle at the top crossing is 55°, the corresponding angle at the lower crossing is also 55°; the angle on a straight line next to that is 180 − 55 = 125°, and the angle vertically opposite it is 55° again.

With algebra, write the fact as an equation. Alternate angles (3x + 10)° and (x + 50)° are equal, so 3x + 10 = x + 50, 2x = 40 and x = 20. Co-interior angles 2x and (x + 30)° add to 180°, so 3x + 30 = 180 and x = 50.

A triangle drawn between two parallel lines still has angles adding to 180°, so its third angle is 180 minus the other two.