supplementary angles

Two angles are called supplementary angles  of each other if the sum of their measures (http://planetmath.org/AngleMeasure) is equal to the straight angle $\pi$, i.e. (http://planetmath.org/Ie) ${180}^{\circ }$.

For example, when two lines intersect each other, they the plane into four disjoint domains (http://planetmath.org/Domain2) corresponding to four convex angles; then any of these angles has a supplementary angle on either side of it (see linear pair). However, two angles that are supplementary to each other do not need to have a common side — see e.g. (http://planetmath.org/Eg) an entry regarding opposing angles in a cyclic quadrilateral (http://planetmath.org/OpposingAnglesInACyclicQuadrilateralAreSupplementary).

Supplementary angles have always equal sines, but the cosines are opposite numbers:

$$\sin (\pi -\alpha )=\sin \alpha ,\cos (\pi -\alpha )=-\cos \alpha$$

These formulae may be proved by using the subtraction formulas of sine and cosine.