Question

question 4 or 5
select the correct answer.
in the figure, $overrightarrow{pq}paralleloverrightarrow{sr}$, $overrightarrow{ps}perpoverrightarrow{pq}$ and $overrightarrow{sr}$, and $overrightarrow{qr}perpoverrightarrow{pq}$ and $overrightarrow{sr}$.
which statement describes a relationship between angles in the figure?
$mangle sru = mangle pqt\times2$
$mangle pqr = mangle tqrdiv2$
$mangle pqr = mangle psrdiv2$
$mangle sru = mangle qps\times2$

Explanation:

Step1: Analyze angle - properties

Since $\overrightarrow{PQ}\parallel\overrightarrow{SR}$, $\angle PQR$ and $\angle SRU$ are corresponding angles, so $\angle PQR=\angle SRU$. Also, $\angle QPS = 90^{\circ}$ and $\angle SRU = 90^{\circ}$ (because $PS\perp PQ$, $SR\perp PQ$, and $\overrightarrow{PQ}\parallel\overrightarrow{SR}$).

Step2: Check each option

• Option 1: $\angle SRU = 90^{\circ}$, $\angle PQT$ is not $45^{\circ}$ generally, so $m\angle SRU

eq m\angle PQT\times2$.

• Option 2: $\angle PQR = 90^{\circ}$, $\angle TQR = 90^{\circ}$, $m\angle PQR

eq m\angle TQR\div2$.

• Option 3: $\angle PQR = 90^{\circ}$, $\angle PSR = 90^{\circ}$, $m\angle PQR

eq m\angle PSR\div2$.

• Option 4: $\angle SRU = 90^{\circ}$, $\angle QPS = 90^{\circ}$, so $m\angle SRU=m\angle QPS$.

Answer:

$m\angle SRU = m\angle QPS\times2$ is the correct relationship as $\angle SRU = 90^{\circ}$ and $\angle QPS = 90^{\circ}$