Sum of angles of a triangle Sum of angles of a triangle In several geometries, a triangle has three "vertices" and three "sides", where three angles of a triangle are formed at each vertex by a pair of adjacent sides. In a Euclidean space, the sum of measures of these three angles of any triangle is invariably equal to the straight angle, also expressed as 180 °, radians, two right angles, or a half-turn. It was unknown for a long time whether other geometries exist, where this sum is different. The influence of this problem on mathematics was particularly strong during the 19th century. Ultimately, the
Proofs of trigonometric identities division identity They remain valid for angles greater than 90° and for negative angles. Or Two angles whose sum is π/2 radians (90 degrees) are "complementary". In the diagram, the angles at vertices A and B are complementary, so we can exchange a and b, and change θ to π/2 − θ, obtaining: Identity 1: The following two results follow from this and the ratio identities. To obtain the first, divide both sides of formula_21 by formula_23; for the second, divide by formula_24. Similarly Identity 2: The following accounts for all three reciprocal functions. Proof 2: Refer to the triangle