Question 17·Medium·Lines, Angles, and Triangles

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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 $m\angle Q$ ?

When you see a triangle with a side extended to form an angle outside the triangle, immediately think of the exterior angle theorem: the exterior angle equals the sum of the two remote interior angles. Identify which interior angles are remote (not adjacent to the exterior angle), write a simple equation (exterior angle = interior angle + interior angle), substitute the given values, and do one quick subtraction to solve—this is faster and less error-prone than trying to find all three interior angles separately.

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Classify angle PRS

Side $\overline{QR}$ is extended through $R$ to $S$ . What kind of angle is $\angle PRS$ relative to triangle $\triangle PQR$ ?

Recall a key theorem about exterior angles

For a triangle, how is an exterior angle related to the two interior angles that are not next to it?

Set up an equation

Write an equation relating $m\angle PRS$ to $m\angle P$ and $m\angle Q$ , then substitute $130^\circ$ for $m\angle PRS$ and $45^\circ$ for $m\angle P$ .

Isolate angle Q

After you substitute, isolate $m\angle Q$ by doing the same operation on both sides of the equation.

Desmos Guide

Use Desmos to solve the equation

In Desmos, type the expression 130 - 45. The numerical result shown is the value of $m\angle Q$ in degrees.

Step-by-step Explanation

Identify the type of angle at R

Angle $\angle PRS$ is formed by extending side $\overline{QR}$ past $R$ , so it is an exterior angle of triangle $\triangle PQR$ at vertex $R$ .

Use the exterior angle theorem

For any triangle, an exterior angle equals the sum of the two remote interior angles (the two angles inside the triangle that are not adjacent to the exterior angle).

Here, the remote interior angles to exterior angle $\angle PRS$ are $\angle P$ and $\angle Q$ . So we can write:

m\angle PRS = m\angle P + m\angle Q

Substitute the known angle measures:

130 = 45 + m\angle Q

Solve for the measure of angle Q

Solve the equation from the previous step:

\begin{aligned} 130 &= 45 + m\angle Q \\ 130 - 45 &= m\angle Q \\ 85 &= m\angle Q \end{aligned}

So $m\angle Q = 85^\circ$ , which corresponds to choice C.

Question 17·Medium·Lines, Angles, and Triangles

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation