In , the measure of an exterior angle at vertex is . The measure of is . What is the measure of , in degrees? (Disregard the degree symbol when entering your answer.)

Solution available with step-by-step explanation

SAT Lines, Angles & Triangles Question 47 | Free SAT Prep | Aniko

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Question 47·Medium·Lines, Angles, and Triangles

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In $\triangle PQR$ , the measure of an exterior angle at vertex $R$ is $138^\circ$ . The measure of $\angle P$ is $46^\circ$ . What is the measure of $\angle Q$ , in degrees? (Disregard the degree symbol when entering your answer.)

Your answer

(Express the answer as an integer)

For triangle problems involving an exterior angle, immediately recall the exterior angle theorem: the exterior angle equals the sum of the two non-adjacent interior angles. Identify those two remote interior angles, set up a simple linear equation (exterior angle = given interior angle + unknown interior angle), and solve with one step of subtraction. This avoids unnecessary work and helps you move quickly through similar geometry questions on the SAT.

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Think about the relationship between an exterior angle and interior angles

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

Identify which angles are remote interior angles

The exterior angle is at vertex $R$ . Which two interior angles of $\triangle PQR$ are not adjacent to this exterior angle?

Translate the relationship into an equation

Once you know that the exterior angle equals the sum of the two remote interior angles, write an equation involving $138$ , $46$ , and the unknown measure of $\angle Q$ , then solve for $\angle Q$ .

Desmos Guide

Model the equation in Desmos

In Desmos, enter the two equations y = x + 46 and y = 138. These represent the relationship $138 = 46 + x$ between the exterior angle and the two remote interior angles, where $x$ is the measure of $\angle Q$ .

Find the intersection to get angle Q

Click or tap on the point where the two lines intersect. The x-coordinate of this intersection point is the solution to the equation, which is the measure of $\angle Q$ in degrees.

Step-by-step Explanation

Recall the exterior angle theorem

For any triangle, the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles (the two interior angles that are not adjacent to the exterior angle).

In $\triangle PQR$ , the exterior angle is at vertex $R$ , so the remote interior angles are $\angle P$ and $\angle Q$ .

Set up the equation using the given angles

We are told:

The exterior angle at $R$ is $138^\circ$ .
$\angle P = 46^\circ$ .

By the exterior angle theorem:

\text{exterior angle at }R = \angle P + \angle Q

Substitute the given values:

138 = 46 + \angle Q

Solve for the measure of angle Q

Solve the equation from the previous step:

138 = 46 + \angle Q

Subtract $46$ from both sides:

138 - 46 = \angle Q

92 = \angle Q

So the measure of $\angle Q$ is $92^\circ$ , and you would enter $92$ as your answer.

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation