Question 336·Easy·Math

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A stage technician is aiming two spotlights whose beams form straight lines on a diagram. In the figure, lines $m$ and $n$ intersect.

The angle between the upward ray of line $n$ and the rightward ray of line $m$ measures $132^\circ$ . The angle between the upward ray of line $n$ and the leftward ray of line $m$ measures $x^\circ$ .

What is the value of $x$ ?

At an intersection, check whether the marked angles sit side by side or across from each other. Neighboring angles whose outer rays point in opposite directions form a linear pair and add to $180^\circ$ . Write that angle sum, then use a Desmos regression to find the unknown. Don't set neighboring angles equal; that rule applies to angles across the intersection.

Hints

Hint 1
A linear pair is two neighboring angles that fill a straight angle. The leftward and rightward rays of line $m$ point in opposite directions. What must the two angles between those rays add to?
Hint 2
Angles that share a ray are not vertical angles, which sit across from each other and are equal. Add the two labeled measures and set their sum equal to the straight-angle total.

Step-by-step

Use the straight angle
Step 1
Identify the angle relationship
The $132^\circ$ and $x^\circ$ angles share the upward ray of $n$ . Their other rays point in opposite directions along $m$ , so they form a linear pair: neighboring angles that fill a straight angle add to $180^\circ$ . They are not vertical angles, which sit across from each other and are equal. Use the straight line, not the picture's apparent angle sizes.
Step 2
Write the angle sum
The two labeled angles fill the straight angle on one side of $m$ , so their measures add to $180$ :
$x+132=180.$
Step 3
Find the missing angle
Type $x_1+132\sim180$ in Desmos. Use $x_1$ as the unknown because Desmos treats $x$ as a graph variable; $\sim$ tells it to find a value that makes the two sides equal. Under PARAMETERS, Desmos shows $x_1=48$ . So the angle labeled $x^\circ$ measures $48^\circ$ . Choice B.

Lessons that teach this

SAT Geometry and TrigonometryBeginnerCoreUse line and angle relationships

Question 336·Easy·Math

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