# 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:

$$\mathrm{sin}(\pi -\alpha )=\mathrm{sin}\alpha ,\mathrm{cos}(\pi -\alpha )=-\mathrm{cos}\alpha $$ |

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 |