Question 268·Medium·Math

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In the figure above, $\triangle PQR$ is shown with side $\overline{QR}$ extended through $R$ to point $S$ .

If $m\angle PRS = 130^{\circ}$ and $m\angle P = 45^{\circ}$ , what is $m\angle Q$ ?

A side extended past a triangle makes an exterior angle. The outside angle equals the sum of the two inside angles away from it, so use those two angles to find the missing one. Don’t treat the outside angle as equal to the inside angle beside it; those neighboring angles add to $180^\circ$ . Trust the labels, not how wide the angles look.

Hints

Hint 1
An exterior angle lies outside the triangle where a side continues past a corner. Since $QR$ continues through $R$ to $S$ , identify the outside angle at $R$ before using the marked measure.
Hint 2
The exterior angle equals the two inside angles away from $R$ added together. Those angles are at $P$ and $Q$ . What equation puts their sum equal to the marked outside angle?

Step-by-step

Use the exterior angle
Step 1
Relate the outside angle to the two far angles
No Desmos needed. The exterior-angle fact leaves one subtraction. Because $QR$ extends through $R$ to $S$ , $\angle PRS$ is an exterior angle, outside the triangle. An exterior angle equals the two interior angles away from it added together. Those are the angles at $P$ and $Q$ , so the marked outside angle gives:
$130^\circ=45^\circ+m\angle Q.$
The inside angle at $R$ touches the exterior angle, so it doesn’t belong in this sum.
Step 2
Find the angle at Q
Subtract $45^\circ$ from both sides to leave the requested angle:
$m\angle Q=130^\circ-45^\circ.$
Calculate the difference:
$m\angle Q=85^\circ.$
So the angle at $Q$ measures $85^\circ$ . Choice C.

Question 268·Medium·Math

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