Math / Geometry

Questionnthe given figure, $\overrightarrow{R L} \| \overrightarrow{Q P}$ . Find $\angle P R S$ if $m \angle P Q R=46^{\circ}$ and $m \angle Q P R=65^{\circ}$ .

Studdy Solution

STEP 1

1. The lines $\overrightarrow{RL}$ and $\overrightarrow{QP}$ are parallel.
2. The angles are measured in degrees.
3. The sum of angles in a triangle is $180^\circ$ .

STEP 2

1. Use the triangle angle sum property to find $\angle QRP$ .
2. Use the parallel line property to find $\angle PRS$ .

STEP 3

We know that in triangle $\triangle PQR$ , the sum of the angles is $180^\circ$ . Therefore, we can write:
$m \angle PQR + m \angle QPR + m \angle QRP = 180^\circ$
Substitute the known values:
$46^\circ + 65^\circ + m \angle QRP = 180^\circ$

STEP 4

Solve for $m \angle QRP$ :
$m \angle QRP = 180^\circ - 46^\circ - 65^\circ$ $m \angle QRP = 69^\circ$

STEP 5

Since $\overrightarrow{RL} \| \overrightarrow{QP}$ , by the Alternate Interior Angles Theorem, we know that $\angle QRP = \angle PRS$ .
Therefore, $m \angle PRS = m \angle QRP = 69^\circ$ .
The measure of $\angle PRS$ is:
$\boxed{69^\circ}$

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