Elementary Geometry for College Students (6th Edition)

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

Chapter 3 - Section 3.5 - Inequalities in a Triangle - Exercises - Page 160: 36


Since this is an indirect proof, we always assume the thing that is opposite to what we want to prove is true. Thus: 1. ZW is perpendicular to XY. 2. If ZW is perpendicular to XY, then XYZ is broken into two separate, equal triangles. 3. If there are equal triangles, then two sides of the original triangle XYZ are identical. 4. If two sides are identical to angles are identical. 5. If two angles are identical, then the triangle is not scalene by the definition of a scalene triangle. 6. The triangle is scalene. 7. Thus, this means that the original statement in (1) is false, so ZW is not perpendicular to XY.

Work Step by Step

1-In a scalene triangle XYZ in which $\overline{ZW} bisects \angle XZY $ let us assume that ZW is perpendicular to segment XY. 2- segment ZW separates the $\triangle XYZ $ into two right triangles$ ( \triangle ZWY and \triangle ZWX ) $ 3- these two triangle by our assumption they must be congruent by ASA since $ \angle ZWY = \angle ZWX= 90^{\circ}, \overline{ZW}=\overline{ZW}$ by identity and $\angle YZW = \angle WZX $ by definition of angle bisector. 5- therefore $ \angle Y = \angle X $ by CPCTC which contradicts the hypothesis, also $\overline{YZ}= \overline{XZ} $by CPCTC contradicts the hypothesis that triangle XYZ is a scalene triangle. 6- thus, the assumed statement must be false and segment ZW is not perpendicular to XY.
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