Geometry: Common Core (15th Edition)

Published by Prentice Hall
ISBN 10: 0133281159
ISBN 13: 978-0-13328-115-6

Chapter 5 - Relationships Within Triangles - 5-7 Inequalities in Two Triangles - Lesson Check - Page 336: 4


Her conclusion should be: $m \angle ABD$ > $m \angle CDB$

Work Step by Step

According to the Hinge Theorem, if two consecutive sides of a triangle are congruent to two consecutive sides in another triangle, but their included angles are not congruent, then the side that is opposite to the larger included angle is longer than the side that is opposite the smaller included angle. However, in this diagram, our friend is not talking about the included angle that lies between the two congruent sides. $AD$ in $\triangle ABD$ is $9$ whereas $BC$ in $\triangle CDB$ is only $8$; therefore, $AD$ is the longer side and the angle opposite to it will be the larger angle. Our friend should say that the angle opposite $\overline{AD}$, which is $\angle ABD$, is greater than the angle opposite $\overline{BC}$, which is $\angle CDB$. Her conclusion should be: $m \angle ABD$ > $m \angle CDB$
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