Vertical angles
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It has been suggested that this article or section be merged with Complementary angles , Supplementary angles , Adjacent angles and Transversal (geometry) to Special angle relationships. (Discuss) Proposed since December 2011. |
In geometry, a pair of angles is said to be vertical (also opposite and vertically opposite, which is abbreviated as vert. opp. ∠s[1]) if the angles are formed from two intersecting lines and the angles are not adjacent. The two angles share a vertex. Such angles are equal in measure and can be described as "equal" (in the UK or the USA) or "congruent" (in the USA).[2]
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[edit] Vertical angle theorem
When two straight lines intersect at a point, four angles are made. The non-adjacent angles are called vertical or opposite or vertically opposite angles. Also, each pair of adjacent angles forms a straight line and the two angles are supplementary.[3] Since either of a pair of vertical angles is supplementary to either of the adjacent angles, the vertical angles are equal in measure.
[edit] Algebraic solution for Vertical Angles
In the figure, assume the measure of Angle A = x. When two adjacent angles form a straight line, they are supplementary. Therefore, the measure of Angle C = 180 − x. Similarly, the measure of Angle D = 180 − x. Both Angle C and Angle D have measures equal to 180 - x and are congruent. Since Angle B is supplementary to both Angles C and D, either of these angle measures may be used to determine the measure of Angle B. Using the measure of either Angle C or Angle D we find the measure of Angle B = 180 - (180 - x) = 180 - 180 + x = x. Therefore, both Angle A and Angle B have measures equal to x and are equal in measure.
[edit] References
- ^ Wong, TW; Wong, MS. "Angles in Intersecting and Parallel Lines". New Century Mathematics. 1B (1 ed.). Hong Kong: Oxford University Press. pp. 161–163. ISBN 9780198001768.
- ^ Euclid (c. 300 BC). The Elements. Proposition I:15.
- ^ Euclid (c. 300 BC). The Elements. Proposition I:13.
[edit] External links
- Definition and properties of vertical angles With interactive applet
- Angle definition pages with interactive applets that are also useful in a classroom setting. Math Open Reference
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