Scalene triangle

01 What is always true

∠DAB means the angle at A — the middle letter is always the corner, and the outer two say which sides form it.

• No two sides have to be equal.

  AB = 80, BC = 71.1, CA = 51.5

• All 3 angles add up to 180°.

  True of every 3-sided shape.

  Why is it always 180°?

  1. Tear the three corners off the triangle and lay them side by side, all touching at one point. They fill a straight line — no gap, no overlap.

  2. A straight line is 180°. So the three angles together must come to 180°, because that is exactly the space they filled.

  3. Nothing about this triangle was used. Squash it, stretch it, turn it — the corners still fit the line, because the fit does not depend on the shape. That is why it holds for every triangle there is.

  4. It is also where the quadrilateral's 360° comes from: cut a quadrilateral along one diagonal and you have two triangles, so 180° + 180° = 360°.

02 Why it works

Scalene is what a triangle is when it is nothing more specific. There are no equal pairs to lean on, so a question about one has to give you enough angles or lengths to work from.

The one thing you still get free is the row above: the three angles add to 180°. Given any two of them, the third follows — that holds for every triangle, however irregular.