Elementary Geometry for College Students (6th Edition)

Published by Brooks Cole
ISBN 10: 9781285195698
ISBN 13: 978-1-28519-569-8

Chapter 6 - Section 6.1 - Circles and Related Segments and Angles - Exercises - Page 277: 38


We are told MN and PQ are parallel, so it follows that corresponding angles of the same transversal are also parallel. Therefore, angle MNP is congruent to angle NPQ, and angle MQP is congruent to angle NMQ. In addition, because QMN and NPQ are formed by the same congruent arcs and transversal, they are congruent. Thus, by substitution, it follows that base angles are congruent. Since the base angles are congruent, the trapezoid is isosceles.

Work Step by Step

To show that MNPQ is an isosceles trapezoid, its enough to show the legs are congruent or pair of angle base are congruent or the diagonals are congruent. In our case, given that MN || PQ, we conclude that angle NMP congruent to angle MPQ ( alternate angles are congruent ). By theorem, $ m\angle NMP = 1/2 arc MQ $ and $ m\angle MPQ= 1/2 arc MQ $. Since both angles are congruent then by setting up an equality therefore MQ=NP.
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