Vertical angles

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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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.

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.

References

  1. ^ 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. 
  2. ^ Euclid (c. 300 BC). The Elements.  Proposition I:15.
  3. ^ Euclid (c. 300 BC). The Elements.  Proposition I:13.

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