Angles at parallel lines

Two railway tracks run to the horizon and never meet. Lay a sleeper across them at a slant and angles appear at both tracks, and they are no accident: they repeat. That is exactly what happens when one straight line cuts two parallel lines. Once you spot the repetition, a single measured angle gives you all the others, with no protractor involved. Three pairs of angles do the job, and you will meet all three here.

Eight angles, but only two sizes

Parallel means: two straight lines keep the same distance everywhere and never meet. A third line that cuts both of them is called the transversal. At each of the two crossing points four angles appear, eight in total. At a single crossing point two familiar rules already hold: two angles lying exactly opposite each other are called vertical angles and are equal. Two angles that sit next to each other and together form a straight line are called adjacent angles and add up to 180°. With two parallels the surprise is added: among the eight angles there are only two different sizes. If you measure 65° at the upper crossing point, then every one of the eight angles is either 65° or 115°, because 65° + 115° = 180°. Only when the transversal stands perpendicular to the parallels do both sizes coincide and all eight angles measure 90°.

Corresponding and alternate angles

Corresponding angles sit at exactly the same spot at both crossing points: either both at the top right or both at the bottom left. You recognise them by the shape of a big letter F, which is why they are also called F angles. Because the transversal meets both parallels at the same slant, corresponding angles are always equal. If the angle at the top right of the first crossing point is 65°, then the angle at the top right of the second crossing point measures 65° as well.

Alternate angles lie crosswise: one to the left of the transversal, the other to its right, both between the parallels. Their memory shape is a big letter Z, hence the name Z angles. Alternate angles are always equal too, and that can be justified in two steps. First take the corresponding angle, which is equal, and then its vertical angle, which is equal as well. Equal twice over is still equal, so the alternate angles match.

Co-interior angles make 180°

The third pair also lies between the parallels, but both angles sit on the same side of the transversal. You recognise these co-interior angles by the shape of a big letter U. They are not equal, they add up to 180° instead. The reason: one of them is the adjacent angle of an alternate angle of the other. The alternate angle is equal in size, and adjacent angles always add up to 180°. So if one is 65°, the other one is left with 180° − 65° = 115°.

Finding missing angles

Everything together in one example: a transversal meets two parallels and at the upper crossing point one angle is given as 118°. Its adjacent angle measures 180° − 118° = 62°. The matching corresponding angle at the lower crossing point is 118° again, so is the alternate angle, and the angle that forms a U with the 118° measures 62°. That fills in the whole figure. If two angles are given as terms with $x$ instead of numbers, you do the same: set equal angles equal, for example $2x = x + 30$, and for the U shape set the sum to 180°, for example $2x + x = 180$. Solving the equation gives $x$, and then you know both angles.