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 - Practice and Problem-Solving Exercises - Page 337: 12


$6 < x < 24$

Work Step by Step

According to the Converse of the Hinge Theorem, if two consecutive sides in a triangle are congruent to two consecutive sides in another triangle, but their third sides are not congruent, then the included angle that is opposite the longer side is the larger angle of the included angles in both triangles. Two sides in one triangle are congruent to two consecutive sides in another triangle. The angle opposite to the longer side has the greater value. Let's write an inequality reflecting this information: $36 > 2x - 12$ Let's add $12$ to each side of the inequality and flip the inequality around to simplify: $2x < 48$ Divide each side by $2$: $x < 24$ Also, $2x - 12$ cannot be $0$ or less because you cannot have an angle that is $0^{\circ}$ or less, so let's set up the inequality to find out what the lowest value for $x$ can be: $2x - 12 > 0$ Add $12$ to each side of the equation: $2x > 12$ Divide each side by $2$: $x > 6$ Let's write the range of $x$: $6 < x < 24$
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