Co-interior angles sum to 180
A transversal crossing two parallel lines makes a pair that is not equal, but always sums to a straight angle.
The last two lessons found matching pairs across a transversal. This one finds a pair that does not match, but is no less predictable.
The same figure as the last two lessons: 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, always summing to 180 degrees; dragging P never changes either. This needs JavaScript switched on.
Two angles on the same side of a transversal, one at each crossing, like the marked angle at G and the marked angle at H, are called co-interior angles. “Interior” because both sit in the strip between the two lines; “co-” because they share that same side of the transversal — and unlike the last two lessons’ pairs, this one does not match, it sums to a fixed total instead.
Reaching 180 from alternate angles and a straight line
This needs one new observation and one old fact, the same pairing used for alternate angles in the last lesson, but with a different old fact supplying the second half.
The angle at H in this lesson and the alternate angle at H from the last lesson sit on opposite sides of the transversal at the same crossing, so together they cover the whole straight angle there:
That is exactly the straight-line fact from five lessons ago, applied at H instead of at the point it was first shown with. The lesson on alternate angles already showed
Substituting the second into the first,
which is the fact the drag showed but did not, by itself, prove. This is the third and last pairing found from one figure, completing the set of matching and summing relationships a transversal makes with two parallel lines.
A transversal crossing two parallel lines now has three proved pairs: corresponding and alternate, both equal, and co-interior, which sums to 180 instead. The next lesson turns this around: instead of assuming the lines are parallel and predicting the angles, it asks what a matching pair of angles tells you about whether the lines are parallel at all.
The transversal crosses line AC at G at 72°. What does it read at H, in the co-interior position?
because co-interior angles always sum to 180.
Explain why, using the alternate angle at H as a stepping stone, rather than just citing the co-interior fact directly.
The angle at H in this lesson and the alternate angle at H sit on a straight line together, so they sum to 180 (the straight-line fact from five lessons ago, reapplied at H). The alternate angle at H equals the angle at G (previous lesson). Substituting one into the other gives the angle at G and the co-interior angle at H summing to 180.
P is dragged so the parallel line sits much closer to line AC. Do the two readings, at G and 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.