Geometry: Common Core (15th Edition)

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

Chapter 4 - Congruent Triangles - 4-1 Congruent Figures - Practice and Problem-Solving Exercises - Page 223: 40


$t = 2$ $x = 15$

Work Step by Step

We are given that $\triangle ABC$ is congruent to $\triangle KLM$. We know that corresponding parts of congruent angles are congruent. Therefore, we can deduce that $\overline{AB}$ is congruent to $\overline{KL}$, so let us set those two line segments equal to one another to find the value of $t$: $AB = KL$ $4 = 2t$ Divide each side by $2$ to solve for $t$: $t = 2$ According to the triangle sum theorem, we know that the sum of the three interior angles of a triangle equals $180^{\circ}$. If we are already given two of the angles, we can figure out the third angle. If $\triangle ABC$ is congruent to $\triangle KLM$, then $\angle A$ is congruent to $\angle K$, $\angle B$ is congruent to $\angle L$, and $\angle C$ is congruent to $\angle M$. This means that both $\angle A$ and $\angle K$ are $45^{\circ}$, and $\angle B$ and $\angle L$ are $90^{\circ}$. $m \angle C$ is given by the following equation: $m \angle C = 180 - (45 + 90)$ Evaluate the parentheses first: $m \angle C = 180 - (135)$ Subtract to find $m \angle C$: $m \angle C = 45$ Since $\angle C$ and $\angle M$ are congruent, we can set them equal to one another to solve for $x$: $45 = 3x$ Divide both sides by $3$ to solve for $x$: $x = 15$
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