Question 260·Hard·Math

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Two straight conveyor belts are represented by parallel lines $\ell_1$ and $\ell_2$ . Point $P$ lies on $\ell_1$ , and points $Q$ and $R$ lie on $\ell_2$ . Segments $PQ$ and $PR$ are drawn.

The measure of the angle between $PQ$ and $\ell_1$ (on the interior side between the two parallel lines) is $(3x-20)^\circ$ . The measure of $\angle PQR$ is $(2x+10)^\circ$ , and the measure of $\angle PRQ$ is $(x+40)^\circ$ .

Which choice is the measure of $\angle QPR$ ?

A triangle with one side on a parallel line gives you two angle facts to use. A side crossing both lines is a transversal, and alternate interior angles on opposite sides of it are equal. Use that equality to find the variable with a Desmos regression, then use the triangle’s $180^\circ$ angle sum. Don’t stop at the variable or a base angle when the question asks for the angle at the top.

Hints

Hint 1
Segment $PQ$ crosses both parallel lines. The marked angle at $P$ and the angle at $Q$ are alternate interior angles: they sit between the lines on opposite sides of $PQ$ . What equation does that give you?
Hint 2
A Desmos regression can find the variable in your angle equation. Change each $x$ to $x_1$ and change $=$ to $\sim$ . The value it finds is not yet the requested angle.
Hint 3
The three interior angles of a triangle add to $180^\circ$ . Once you know the angles at $Q$ and $R$ , subtract both from $180^\circ$ to get the angle at $P$ .

Step-by-step

Match parallel-line angles, then use the triangle sum
Step 1
Match the angles made by $PQ$
The marked angle at $P$ and $\angle PQR$ lie between the parallel lines on opposite sides of $PQ$ . They’re alternate interior angles: angles in these positions are equal, so
$3x-20=2x+10.$
Step 2
Find the variable
Type $3x_1-20\sim 2x_1+10$ . In this regression, $x_1$ is the unknown and $\sim$ asks Desmos to match the two sides. Under PARAMETERS, Desmos shows $x_1=30$ . That’s the value of $x$ , not an angle to choose.
Step 3
Find the two angles along $QR$
Type $2x_1+10$ and $x_1+40$ below the regression. Desmos prints $70$ for each. These are the base angles at $Q$ and $R$ , the two ends of $QR$ . Neither is the angle at $P$ .
Step 4
Find the angle at $P$
A triangle’s three interior angles add to $180^\circ$ . Type $180-(2x_1+10)-(x_1+40)$ to subtract the angles at $Q$ and $R$ . Desmos prints $40$ , so the angle at $P$ , $\angle QPR$ , measures $40^\circ$ . Choice A.

Question 260·Hard·Math

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