Angles on a straight line sum to 180
Splitting a straight angle in two, and why the two parts always add back up to a straight angle.
A straight angle, 180°, was named two lessons ago. This lesson asks what happens when something splits one in two.
Here is a straight line through A and C, with B sitting exactly between them. A ray runs from B out to D, splitting the straight angle at B into two smaller ones. Drag D, or swing the whole line by dragging A or C, and watch what the two pieces of the split add up to.
A straight line through A and C, with B exactly between them, and a ray out to D splitting the straight angle in two. A, C and D can all be dragged. This needs JavaScript switched on.
Try dragging D onto the line itself, on the C side: one angle should read 180°, the other 0°, and the total is still 180.
The vocabulary for a split straight angle
B is the vertex the two smaller angles share. The two of them, angle ABD and angle DBC, are called adjacent angles here: they share a vertex and an arm (BD), and together they exactly cover the straight angle ABC. Two angles whose measures add to 180° are supplementary, whether or not they happen to sit on a straight line together — angle ABD and angle DBC are supplementary because they do.
Why the split always totals 180
Angle ABD and angle DBC are read off two entirely separate measurements: one is the turn from BA to BD, the other is the turn from BD to BC. Neither is calculated from the other. What makes them add to 180 every time is a single fact from two lessons ago: BA and BC point in exactly opposite directions, because B is the midpoint of a straight line through A and C.
As D swings from lying along BA to lying along BC, one angle sweeps continuously from 0° up to 180° while the other sweeps from 180° down to 0° — so their total is 180 at every moment in between, not just at the start and end. That argument never mentions which way the line itself points, which is why swinging the whole line changes nothing.
This is the last time the course derives a sum-of-180 fact from the parallel-postulate lesson directly — everything after this reuses this result instead of re-deriving it.
This is the first fact in the course built entirely from two earlier ones (a straight angle is 180°, and directions from a shared point add turn) rather than from a new agreement. The next lesson, angles at a point, is built from this one the same way.
Angle ABD measures 62°. What is angle DBC, given that A, B and C lie on a straight line?
Angle ABD measures 90°. What is angle DBC?
D is dragged so that BD lies exactly along BA. What do angle ABD and angle DBC read?
Angle ABD is 0°, because BD and BA point the same way. Angle DBC is then the whole straight angle:
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.