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(Vertical Angle Theorem) Let A, B, C, D be four distinct points in the plane such that the betweenness relations A∗X∗C and B∗X∗D are valid. Then the measures of the vertical angles satisfy |∠AXB| = |∠CXD|. Proof. Two applications of the Supplemental Angle Identity (the sum of their measures is 180.
Theorem All right angles are congruent. Vertical Angles Theorem Vertical angles are equal in measure Theorem If two congruent angles are supplementary, then each is a right angle. Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Converse of the Angle Bisector Theorem
theorems for which the synthetic arguments are better, and this is one reason for employing a combined synthetic-analytic approach to geometry. To illustrate this point, we shall give a proof of the Vertical Angle Theorem (Theorem II.3.7 in the notes) using vector geometry.
Vertical angles are congruent. a1 c a3 and a2 c a4. The following steps show why the Vertical Angles Theorem is true. 1 a1 and a2 are a linear pair, so a1 and a2 are supplementary. 2 a2 and a3 are a linear pair, so a2 and a3 are supplementary. To review the Congruent Supplements Theorem, see p. 69.
VERTICAL ANGLES THEOREM If two angles are vertical angles, then they are congruent. <1≅<3,<2≅<4
Two intersecting lines form pairs of vertical angles and linear pairs. The Linear Pair Postulate formally states the relationship between linear pairs. You can use this postulate to prove the Vertical Angles Congruence Theorem. Proving the Vertical Angles Congruence Theorem Use the given paragraph proof to write
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