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I know why vertical angles are congruent but I dont know why they must be congruent

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3 Answers 3

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Two intersecting lines form two pair of congruent vertical angles. If the vertical angles of two intersecting lines fail to be congruent, then the two intersecting "lines" must, in fact, fail to be lines...so the "vertical angles" would not, in fact, be "vertical angles", by definition.

Perhaps you'd be interested in viewing a proof of this at the Khan Academy video:

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Recall that if $\angle BAC$ and $\angle BAD$ are supplementary angles, and if $\angle B'A'C'$ and $\angle B'A'D'$ are supplementary angles, and if $\angle BAC\cong\angle B'A'C'$, then also $\angle BAD\cong\angle B'A'D'$. (This is Proposition 9.2 on page 92 of Robin Hartshorne's Geometry: Euclid and Beyond.) A proof may be found here.

Now vertical angles are defined by the opposite rays on the same two lines. Suppose $\alpha$ and $\alpha'$ are vertical angles, hence each supplementary to an angle $\beta$. Since $\beta$ is congruent to itself, the above proposition shows that $\alpha\cong\alpha'$.

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When two straight lines intersect at a point, four angles are made. The non-adjacent angles are called vertical or opposite . Also, each pair of adjacent angles forms a straight line and the two angles are supplementary. Since either of a pair of vertical angles is supplementary to either of the adjacent angles, the vertical angles are equal in value or size.

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