Chapter 10 - Geometry - Chapter Summary, Review, and Test - Review Exercises - Page 680: 16

The values of the angles numbered from 1 to 5 are \[\operatorname{m}\measuredangle 1=90{}^\circ \], \[\operatorname{m}\measuredangle 2=90{}^\circ \], \[\operatorname{m}\measuredangle 3=140{}^\circ \], \[\operatorname{m}\measuredangle 4=40{}^\circ \], and\[\operatorname{m}\measuredangle 5=140{}^\circ \].

In the given figure, there is a right angle triangle intersected by a traversal. The angles formed by a right angle triangle are two acute angles and one with a measure of 90\[{}^\circ \]. Thus, in the given figure angle 1 formed at the base of the right angle triangle has a measure of \[90{}^\circ \]. Hence, the measure of angle 1 is \[\operatorname{m}\measuredangle 1=90{}^\circ \]. The angles that together form a straight line form a supplementary angle. The sum of the supplementary angles is \[180{}^\circ \]. Thus, in the given figure angle 1 and angle 2 are supplementary angles and form a straight line. Hence, the measure of angle 2 is \[180{}^\circ -90{}^\circ =90{}^\circ \]. In the given figure, angle 4 and angle given with a measure of \[40{}^\circ \] are nonadjacent angles and thus called vertical angles. The measure of vertical angles are equal. Hence, the measure of angle 4 is \[\operatorname{m}\measuredangle 4=40{}^\circ \]. In the given figure, angle 5 and the angle 4 with a measure of \[40{}^\circ \]are supplementary angles and form a straight line. Hence, the measure of angle 5 is \[180{}^\circ -40{}^\circ =140{}^\circ \]. In the given figure, angle 5 and angle 3 are nonadjacent angles and thus called vertical angles. The measure of vertical angles are equal. Hence, the measure of angle 3 is \[\operatorname{m}\measuredangle 3=140{}^\circ \].