Chapter 10 - Geometry - Chapter Summary, Review, and Test - Review Exercises - Page 680: 13

The values of other seven angles are \[\operatorname{m}\measuredangle 1=138{}^\circ \], \[\operatorname{m}\measuredangle 2=42{}^\circ \], \[\operatorname{m}\measuredangle 3=138{}^\circ \], \[\operatorname{m}\measuredangle 4=138{}^\circ \], \[\operatorname{m}\measuredangle 5=42{}^\circ \], \[\operatorname{m}\measuredangle 6=42{}^\circ \], and \[\operatorname{m}\measuredangle 7=138{}^\circ \].

In the given figure, there are two parallel lines that are intersected by a traversal. The angles formed by nonadjacent angles are called vertical angles. Thus, in the given figure angle 2 and angle given with a measure of \[42{}^\circ \]are vertical angles. The measure of vertical angles are equal. Hence, the measure of angle 2 is \[\operatorname{m}\measuredangle 2=42{}^\circ \]. 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 3 and the angle 2 given with a measure of \[42{}^\circ \] are supplementary angles and form a straight line. Hence, the measure of angle 3 is \[\operatorname{m}\measuredangle 3=180{}^\circ -42{}^\circ =138{}^\circ \]. In the given figure, angle 3 and angle 1 are nonadjacent angles and thus called vertical angles. The measure of vertical angles are equal. Hence, the measure of angle 1 is \[\operatorname{m}\measuredangle 1=138{}^\circ \]. The angles formed by the pair of angles that are on the inner and opposite side of the traversal are called alternate interior angles. Thus, in the given figure angle 5 and angle given are alternate interior angle that are always of the same measure. Hence, the measure of angle 5 is \[\operatorname{m}\measuredangle 5=42{}^\circ \]. In the same manner, angle 4 and angle 3 are alternate interior angles that are always of the same measure. Hence, the measure of angle 4 is \[\operatorname{m}\measuredangle 4=138{}^\circ \]. If there are two parallel lines, the angles formed in the same corners or one angle is interior and one angle is exterior which is on the same side of the traversal are called corresponding angles. Thus, in the given figure angle 6 and angle given are corresponding angles that are always of the same measure. Hence, the measure of angle 6 is \[\operatorname{m}\measuredangle 6=42{}^\circ \]. In the given figure, angle 7 and the angle 6 with a measure of \[42{}^\circ \] are supplementary angles and form a straight line. Hence, the measure of angle 7 is \[\operatorname{m}\measuredangle 7=180{}^\circ -42{}^\circ =138{}^\circ \]. Hence, the values of seven angles are \[\operatorname{m}\measuredangle 1=138{}^\circ \], \[\operatorname{m}\measuredangle 2=42{}^\circ \], \[\operatorname{m}\measuredangle 3=138{}^\circ \], \[\operatorname{m}\measuredangle 4=138{}^\circ \],\[\operatorname{m}\measuredangle 5=42{}^\circ \], \[\operatorname{m}\measuredangle 6=42{}^\circ \], and \[\operatorname{m}\measuredangle 7=138{}^\circ \].