A Tale of Three Intersecting Lines
The angle sum property
Discover what is always true about the three angles of any triangle.
The same trick works for every triangle. Draw XY through A parallel to BC. By alternate angles, ∠B = ∠XAB and ∠C = ∠YAC.
So ∠A + ∠B + ∠C = ∠A + ∠XAB + ∠YAC. Those three angles lie along the straight line XY, so they add to 180°.
The three angles of any triangle add up to 180°. This is the angle sum property.
How did we find it? At first there was no clear link between the angles. Then one clever idea, a parallel line through the top corner, made it obvious. That idea appears in *The Elements*, one of the most important maths books ever, by the Greek mathematician Euclid (around 300 BCE). You can also check it by tearing or folding the three corners of a paper triangle so they sit together on a straight line.
Exterior angle: extend side BC beyond C to a point D. The angle ∠ACD, between the extended side and side AC, is an exterior angle.
If ∠A = 50° and ∠B = 60°: then ∠ACB = 180° − 110° = 70°, and ∠ACD = 180° − 70° = 110°. Notice 110° = 50° + 60°! It always works: an exterior angle equals the sum of the two opposite interior angles (the two that are not next to it).
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