We can first apply the Vertical Angles Theorem, which states that opposite angles in a pair of intersecting lines are congruent. In other words, the angles vertical from each other are congruent. This is the case with angles POQ and ROS.
Let’s use this theorem to create an equation and solve for “x.” We know that (x+21)°=147° because they’re vertical angles.
(x+21)°=147°
Remove the parenthesis and cancel the degree sign for now:
x+21=147
Solve for “x” - subtract 21 from both sides:
x=147-21
x=126
Answer 1: x=126.
Now, let’s solve for the measure of angle POS. We know that angles ROS and POS lie on a straight line. By definition, straight lines measure 180°. Therefore, since angles ROS and POS are part of angle POR (which is a straight line), their sum should be 180°.
Let’s form and solve an equation for m∠POS:
m∠POS+147°=180°
Subtract 147 from both sides:
m∠POS=180-147
m∠POS=33°
Answer 2: m∠POS=33°