Transversals; corresponding angles
A line crossing two parallel lines meets them at the same angle, and here is why.
This is the first lesson about two parallel lines together, rather than one line on its own.
Here is a line through A and C, and a second line through P that runs the same way as the first, however P is dragged: the parallel postulate again, from a few lessons ago. A third line, through E and F, crosses both.
A line through A and C, a second line through P running the same way as the first, and a transversal through E and F crossing both. Dragging P slides the second line along without changing its direction; dragging E or F swings the transversal and changes the angle both lines make with it. This needs JavaScript switched on.
A line that crosses two others like the third one here is a transversal. Where a transversal crosses two parallel lines, the two angles that sit in matching corners, one at each crossing, are called corresponding angles — the pair E and F keep locked together no matter how the transversal is swung.
Why the reading refuses to move
The angle a line makes with the transversal is decided by the line’s direction, not by which point on it you happen to be standing at. The second line is built to always share the first line’s direction exactly, whatever P is. Two lines with the same direction meet any given transversal at the same angle, because “the angle with the transversal” is a fact about direction alone. Sliding P moves where the second line crosses the transversal, not which way the second line runs, so the angle it makes cannot change.
Every parallel-lines lesson still to come rests on this one, and nothing new had to be assumed to reach it.
The next lesson, alternate angles, compares a corresponding angle at one crossing to a vertically opposite angle at the other. This lesson supplies the first half of that comparison; the fact that vertically opposite angles are equal, from two lessons ago, supplies the second.
A transversal crosses two parallel lines. It meets the first at 52°. What angle does it make with the second?
because both lines share the same direction, and the angle with a transversal depends only on direction.
The same transversal is swung to a new direction, so it now meets the first line at 71° instead. What does it read at the second line, without dragging anything else?
because the two lines still share the same direction as each other; only the transversal changed.
A point P slides the second line closer to the first, without turning it. Does either angle reading change?
No. Sliding P changes where the second line sits, not which way it runs, and the angle with the transversal depends only on the direction.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.