Elementary Geometry for College Students (5th Edition)

Published by Brooks Cole
ISBN 10: 1439047901
ISBN 13: 978-1-43904-790-3

Chapter 2 - Section 2.4 - The Angles of a Triangle - Exercises - Page 97: 15


m$\angle2$ = 57.7$^{\circ}$ m$\angle4$ = 41.5$^{\circ}$ m$\angle3$ = 80.8$^{\circ}$

Work Step by Step

$\angle1$ and $\angle2$ are supplementary angles, which means m$\angle1$ + m$\angle2$ must equal 180$^{\circ}$ m$\angle1$ = 122.3$^{\circ}$ + m$\angle2$ = 180$^{\circ}$ Subtract m$\angle1$ from 180$^{\circ}$ to get m$\angle2$ 180$^{\circ}$ - 122.3$^{\circ}$ = 57.7$^{\circ}$ Therefore, m$\angle2$ = 57.7$^{\circ}$ Since j $\parallel$ k we know that $\angle5$ and $\angle4$ are alternate interior angles. Based off of the law of alternate interior angles know that since $\angle5$ and $\angle4$ are alternate interior angles, they are congruent. So m$\angle5$ = m$\angle4$. m$\angle5$ = 41.5$^{\circ}$ Therefore, m$\angle4$ = 41.5$^{\circ}$ Since we know the measures of two of the three $\angle$s which make up $\triangle ABC$ (m$\angle4$ & m$\angle2$) we can figure out the measure of the unknown angle, m$\angle3$. We can find this out because of the properties of the interior angles of a triangle. One of the properties of interior angles of a triangle is that they must add up to 180$^{\circ}$. We know two of the interior angles measurements of $\triangle ABC$ (m$\angle4$ & m$\angle2$) we know the measure of the missing angle must bring their sum to equal 180$^{\circ}$. m$\angle2$ + m$\angle4$ + m$\angle3$ = 180$^{\circ}$ 57.7$^{\circ}$ + 41.5$^{\circ}$ + m$\angle3$ = 180$^{\circ}$ Subtract 99.2$^{\circ}$ from 180$^{\circ}$ Threrefore, m$\angle3$ = 80.8$^{\circ}$
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