Question

1. using properties of parallel lines use the given information to find the measures of the other labeled angles in the figure. for each angle, tell which postulate or theorem you used. given ▸ $overline{pq}paralleloverline{rs}$, $overline{lm}perpoverline{nk}$, $mangle1 = 48^{circ}$

Explanation:

Step1: Vertical - angles are equal

Since $\angle1$ and $\angle3$ are vertical - angles, $m\angle3=m\angle1 = 48^{\circ}$ (Vertical Angles Theorem).

Step2: Supplementary angles

$\angle1$ and $\angle2$ are supplementary. So $m\angle2=180^{\circ}-m\angle1=180 - 48=132^{\circ}$ (Linear - Pair Postulate). Since $\angle2$ and $\angle4$ are vertical - angles, $m\angle4=m\angle2 = 132^{\circ}$ (Vertical Angles Theorem).

Step3: Corresponding angles

Because $\overline{PQ}\parallel\overline{RS}$, $\angle1$ and $\angle6$ are corresponding angles. So $m\angle6=m\angle1 = 48^{\circ}$ (Corresponding Angles Postulate).

Step4: Vertical - angles and supplementary angles

$\angle6$ and $\angle7$ are supplementary. So $m\angle7=180^{\circ}-m\angle6=180 - 48 = 132^{\circ}$ (Linear - Pair Postulate). $\angle4$ and $\angle5$ are vertical - angles, so $m\angle5=m\angle4 = 132^{\circ}$ (Vertical Angles Theorem). Also, since $\overline{LM}\perp\overline{NK}$, the angles formed by their intersection are right - angles.

Answer:

$m\angle2 = 132^{\circ}$ (Linear - Pair Postulate), $m\angle3 = 48^{\circ}$ (Vertical Angles Theorem), $m\angle4 = 132^{\circ}$ (Vertical Angles Theorem), $m\angle5 = 132^{\circ}$ (Vertical Angles Theorem), $m\angle6 = 48^{\circ}$ (Corresponding Angles Postulate), $m\angle7 = 132^{\circ}$ (Linear - Pair Postulate)