Question

m∠opq = m∠por because they are alternate interior angles. m∠pqo = m∠qos because they are alternate interior angles. m∠por + x°+ m∠qos = 180° because the three angles form a straight line. so the value of x must be 50

Explanation:

Step1: Identify equal - angle relationships

Since $\angle OPQ$ and $\angle POR$ are alternate - interior angles, $m\angle OPQ=m\angle POR = 45^{\circ}$. Also, since $\angle PQO$ and $\angle QOS$ are alternate - interior angles, $m\angle PQO=m\angle QOS = 85^{\circ}$.

Step2: Use the straight - line angle sum property

The angles $\angle POR$, $x^{\circ}$, and $\angle QOS$ form a straight line. The sum of angles on a straight line is $180^{\circ}$. So, we have the equation $m\angle POR+x + m\angle QOS=180$. Substituting the known values: $45 + x+85 = 180$.

Step3: Solve for $x$

Combining like terms, we get $130 + x=180$. Then, subtracting 130 from both sides of the equation: $x=180 - 130$. So, $x = 50$.

Answer:

$50$