Triangles & Angle Sums

The three angles of any triangle add to 180°. Any triangle at all — big, small, stretched.

That means knowing two angles always gives you the third by subtraction.

Triangles are named by sides (equilateral: all three equal; isosceles: two equal; scalene: none equal) or by angles (acute, right, obtuse). In an isosceles triangle the angles opposite the equal sides are also equal, which is often the missing step in a problem.

For bigger shapes: split the polygon into triangles. A quadrilateral splits into 2, so 360°. A pentagon splits into 3, so 540°.

The 180° sum follows from the parallel postulate. Draw a line through one vertex parallel to the opposite side; the two alternate interior angle pairs relocate the other two angles onto a straight line at that vertex, and a straight line is 180°.

For a convex n-gon, drawing all diagonals from a single vertex produces n − 2 triangles, giving an interior angle sum of (n − 2)·180°. The exterior angles behave more simply: they sum to 360° for any convex polygon regardless of n, since walking the perimeter once turns you through one full revolution.

The exterior angle theorem for triangles is worth knowing separately: an exterior angle equals the sum of the two non-adjacent interior angles, a direct consequence of both the linear pair and the 180° sum.