The angle sum of a triangle is 180
The first big payoff of the parallel postulate — and the first lesson to spend the construction tool the last one just built.
Every triangle drawn so far, in every textbook a student has ever opened, has had angles summing to 180 degrees. This lesson is not about noticing that pattern again. It is about finding out why it is true — for every triangle, not just the ones that happen to have been drawn.
A triangle, A, B and C, all three draggable into any shape at all. Through A runs a second line, built to always stay parallel to BC, however the triangle changes.
A triangle, A, B and C, all three draggable, with a line through A built to stay parallel to BC however the triangle changes shape. The angle at A that this parallel line reads on the B side always matches the angle at B, and the angle it reads on the C side always matches the angle at C, so the three angles at A, together, always sum to 180 degrees, whatever shape the triangle is dragged into. This needs JavaScript switched on.
The parallel line through A is what makes that total provable rather than merely observed: watch its reading on the B side track the angle at B exactly, and its reading on the C side track the angle at C exactly, however the triangle moves.
An extra line the problem never asked for
A line added to a figure, not because the problem stated it, but because it is the one addition that makes the proof possible, is called an auxiliary line. The triangle itself never needed a fourth line — the proof did. Choosing the right auxiliary line is most of what makes a construction problem hard, and this lesson’s version of that choice was made for you: parallel to one side, through the opposite vertex.
Turning that line into a proof
The auxiliary line through A is parallelLine('A', lineBC): built to share BC’s direction exactly,
whatever that direction currently is. Because AB and AC already cross both BC and this new parallel
line, the last lesson’s construction tool applies twice over, at the same point: the angle between
AB and the parallel line, on the side towards B, is the alternate angle to the angle at B, so it
equals the angle at B; the angle between AC and the parallel line, on the side towards C, is the
alternate angle to the angle at C, so it equals the angle at C.
Those two copies, plus the angle at A itself, are three angles sitting side by side along one straight line — the auxiliary line, straight at A by construction. Angles on a straight line sum to 180, so the copy of angle B, plus angle A, plus the copy of angle C, sum to 180. But the copies equal the originals, so angle B plus angle A plus angle C sum to 180 too. That is the whole theorem.
Every step here is something this course has already earned separately, and none of them assumed the answer. “Parallel lines make alternate angles equal” needed the parallel postulate. “Angles on a straight line sum to 180” needed nothing but the definition of a straight line. Put those two true things next to each other at the same point, and a triangle’s angle sum falls out as a consequence, not a new assumption bolted on. That is also why it is not automatic: a geometry that dropped the parallel postulate would keep the straight-line fact, lose the alternate-angle fact, and lose this theorem with it. Triangles summing to 180 is not a property of triangles. It is a property of the parallel postulate, wearing a triangle’s shape.
This is the first lesson in the course where a construction from an earlier lesson gets reused rather than only practised. The exterior angle theorem and the angle sum of any polygon, both coming next, reuse this result rather than repeating this proof.
Why does the auxiliary line have to be parallel to BC specifically, rather than any other line through A?
Because a line through A in some other direction would not make AB and AC’s alternate angles equal to angle B and angle C. The parallel condition is exactly what the last lesson’s theorem needs to fire; without it, the two copied angles at A would just be two more unknowns, not equal to anything already measured.
Could the same proof be run through vertex B or C instead of A?
Yes — nothing about the argument singles out A. A line through B parallel to AC, or through C parallel to AB, would produce the same three angles side by side along a different straight line, and the same sum. The choice of vertex is bookkeeping, not mathematics.
If a triangle’s three angles are 72, 65 and 43 degrees, is that enough to know it is a valid triangle?
It is enough to know the angles are consistent with one, since they sum to 180 — a necessary condition this lesson just proved. It is not enough to know its side lengths, since infinitely many similarly-shaped triangles, all different sizes, share those same three angles.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.