Elementary Geometry for College Students (6th Edition)

Published by Brooks Cole
ISBN 10:
ISBN 13:

Chapter 4 - Section 4.2 - The Parallelogram and Kite - Exercises - Page 186: 27

Answer

Consider a kite situated vertically: 1. If the diagonal is a perpendicular bisector, then all of the angles next to the bisector are 90 degrees, meaning that they are all congruent. This only happens when there are perpendicular bisectors. 2. The kite creates two triangles. The bottom two triangles share a side, which is congruent to itself by the identity property. 3. The other two sides of the bottom two triangles are congruent only if it is a bisector, so these are congruent given that it is a perpendicular bisector. 4. This forms congruent triangles by SAS. 5. Thus, the corresponding sides are congruent if and only if the diagonals are a perpendicular bisector. 6. Since kites have to have congruent corresponding sides, then the diagonals of kites must be perpendicular bisectors.
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