# supplementary angles

Two angles are called   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:

 $\sin(\pi\!-\!\alpha)\;=\;\sin\alpha,\qquad\cos(\pi\!-\!\alpha)\;=\;-\cos\alpha$

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

 Title supplementary angles Canonical name SupplementaryAngles Date of creation 2013-03-22 17:34:59 Last modified on 2013-03-22 17:34:59 Owner pahio (2872) Last modified by pahio (2872) Numerical id 8 Author pahio (2872) Entry type Definition Classification msc 51M04 Classification msc 51F20 Synonym supplementary Related topic Supplement Related topic Angle Related topic ComplementaryAngles Related topic GoniometricFormulae