Elementary Geometry for College Students (6th Edition)

Published by Brooks Cole
ISBN 10:
ISBN 13:

Chapter 3 - Section 3.3 - Isosceles Triangles - Exercises - Page 147: 42

Answer

Triangle PMN; Triangle MPB; Triangle PAN; Triangle MQN; Triangle MBN; Triangle ANM

Work Step by Step

First of all, triangle PMN is isosceles, for we are told that PM and PN are congruent. Also, by the definition of a bisector, it follows that MQN is isosceles, for MQ is congruent to QN. We now find the angles to find more isosceles triangles, for if base angles are congruent, the triangles are isosceles. Since P is 36 degrees, we find that angle M and angle N are both 72 degrees. This means that angles PMQ and PNQ are 36 degrees, for they are bisected. Since this is the same measure as angle P, this means that triangles MPB and PAN are isosceles. Since angles PMQ and PNQ are 36 degrees, this means that angles NQB and MQN are also 36 degrees, for they are bisected. This means that angle B and angle A are 72 degrees, making triangles MBN and ANM also isosceles.
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