Question

1. in the diagram shown, $overline{qw}$ and $overline{sv}$ intersect at point $r$ and $overline{rt}$ is drawn such that $angle srt=angle wrt$. (a) name a linear pair of angles from the diagram. (b) name a pair of vertical angles from the diagram. (c) if $mangle srt = 70^{circ}$, then find the measure of $angle qrv$ and the measure of $angle qrs$.

Explanation:

Step1: Recall linear - pair property

A linear - pair of angles are adjacent and supplementary. $\angle SRT$ and $\angle SRQ$ form a linear - pair as they are adjacent and $\angle SRT+\angle SRQ = 180^{\circ}$.

Step2: Recall vertical - angles property

Vertical angles are opposite angles formed by the intersection of two lines. $\angle SRT$ and $\angle WRV$ are vertical angles (also $\angle SRQ$ and $\angle VRW$ are vertical angles).

Step3: Find $\angle QRV$

Since $\angle SRT=\angle WRV = 70^{\circ}$ (vertical angles are equal) and $\angle QRV$ and $\angle WRV$ are supplementary ($\angle QRV+\angle WRV = 180^{\circ}$), then $\angle QRV=180^{\circ}-\angle WRV$. Substituting $\angle WRV = 70^{\circ}$, we get $\angle QRV = 180 - 70=110^{\circ}$.

Step4: Find $\angle QRS$

Since $\angle SRT$ and $\angle SRQ$ are a linear - pair and $\angle SRT = 70^{\circ}$, then $\angle SRQ=180^{\circ}-\angle SRT = 180 - 70 = 110^{\circ}$. And $\angle QRS = 110^{\circ}$.

Answer:

(a) $\angle SRT$ and $\angle SRQ$
(b) $\angle SRT$ and $\angle WRV$ (or $\angle SRQ$ and $\angle VRW$)
(c) $\angle QRV = 110^{\circ}$, $\angle QRS = 110^{\circ}$