Parallel Lines Cut by a Transversal

Two parallel lines are identical in direction, so a transversal crossing both hits them at the exact same tilt. That single fact explains all eight angles. At each crossing you get four angles, two sizes repeated, call them the small angle and its supplement (large angle). Slide one crossing down onto the other and the pattern matches up exactly: corresponding angles land on top of each other and are equal. Alternate interior angles sit on opposite sides of the transversal, between the two lines, and are also equal, since the slide plus a half-turn carries one onto the other. Same-side interior angles sit on the same side between the lines; they add to 180 degrees because one is the small angle and the other is the large one at its own crossing.

Once you spot which relationship a pair of angles has, the arithmetic is quick. If one interior angle is 63 degrees, its vertical angle is also 63, its corresponding and alternate angles are 63, and every angle supplementary to it is 117. For algebra versions, alternate interior (or corresponding) angles give an equation like 3x + 15 = 5x - 25, which solves to x = 20; same-side interior angles instead get set equal to 180 minus each other, or summed to 180.

All of this depends on the two lines actually being parallel. If they are not, a transversal still makes two crossings with equal vertical angle pairs at each crossing separately, but corresponding, alternate, and same-side rules break since the lines meet the transversal at different tilts. This same idea of equal tilt reappears in coordinate geometry: parallel lines in the xy-plane always have equal slopes, which is really the same statement about direction made algebraic.