Alternate angles are equal
A transversal crossing two parallel lines makes a second matching pair, on opposite sides.
The last lesson found one matching pair across a transversal. This one finds a second, in a different arrangement, using nothing but the last two lessons.
The same figure as last time: a line through A and C, a second through P running the same way, and a transversal through E and F crossing both, at G and at H.
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, at G and at H. Dragging E or F swings the transversal and changes the angle at G and the angle at H together; dragging P never changes either. This needs JavaScript switched on.
Two angles on opposite sides of a transversal, one at each crossing, like the marked angle at G and the marked angle at H, are called alternate angles. That this pair matches is not a new fact to take on faith — it falls straight out of the last two lessons, put together.
Reaching it from corresponding and vertically opposite angles
This does not need a new idea, only the last two lessons, used together at H:
The last lesson showed that the angle at G and the corresponding angle at H, in the matching position, always read the same number:
Two lessons ago showed that vertically opposite angles are equal. The marked angle at H in this lesson and the corresponding angle at H from the last lesson are exactly that: two angles directly across the crossing H from each other. So
Put the two together, and the corresponding angle at H cancels out of the middle, leaving
which is the fact the drag showed but did not, by itself, prove. This is the first fact in the course built from two different earlier facts joined together, rather than one fact used twice on itself: neither half alone reaches an alternate pair, but joined, they do.
A transversal crossing two parallel lines now has two proved matching pairs: corresponding (last lesson) and alternate (this one). The next lesson, co-interior angles, finds the pair that is not equal, but sums to a known total instead — the straight-line fact from three lessons ago, reapplied at H the same way vertically opposite angles were reapplied here.
The transversal crosses line AC at G at 58°. What does it read at H, in the alternate position?
because the angle at G and the alternate angle at H are always equal.
Explain why, using the corresponding angle at H as a stepping stone, rather than just citing the alternate-angles fact directly.
The corresponding angle at H equals the angle at G (previous lesson). The alternate angle at H is vertically opposite to that corresponding angle, so it equals it too (the lesson before that). Both routes reach the same corresponding angle at H, so the angle at G and the alternate angle at H must be equal.
P is dragged so the parallel line sits much further from line AC. Does either reading at G or at H change?
No. Sliding P changes where the second line sits, not which way it runs, and both readings depend only on the direction of the two lines and the transversal, never on P.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.