In triangle , an exterior angle at vertex measures , and one of the remote interior angles measures . What is the measure, in degrees, of the other remote interior angle?

Solution available with step-by-step explanation

SAT Lines, Angles & Triangles Question 49 | Free SAT Prep | Aniko

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Question 49·Easy·Lines, Angles, and Triangles

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In triangle $PQR$ , an exterior angle at vertex $Q$ measures $110^\circ$ , and one of the remote interior angles measures $45^\circ$ . What is the measure, in degrees, of the other remote interior angle?

For exterior-angle triangle questions, immediately recall that an exterior angle equals the sum of the two remote interior angles. Identify which angles are remote (the ones not adjacent to the exterior angle), set up a simple equation like $\text{exterior} = \text{remote}_1 + \text{remote}_2$ , and then solve for the unknown with basic addition or subtraction. Keep the arithmetic clean to avoid small subtraction errors that lead to tempting wrong choices.

Hints

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Recall the key triangle fact

Think about the relationship between an exterior angle of a triangle and the two interior angles that are not touching that exterior angle.

Write an equation

Let $x$ be the measure of the unknown remote interior angle. How can you write an equation relating $110$ , $45$ , and $x$ using the exterior angle theorem?

Solve for the unknown angle

Once you have the equation involving $110$ , $45$ , and $x$ , rearrange it to isolate $x$ and compute the result.

Desmos Guide

Use Desmos to compute the missing angle

In Desmos, type 110 - 45 into the expression line. The value that Desmos outputs is the measure of the other remote interior angle.

Step-by-step Explanation

Use the exterior angle theorem

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

So, if the exterior angle at $Q$ is $110^\circ$ and the two remote interior angles are $45^\circ$ and $x$ , then they satisfy:

110 = 45 + x

Solve the equation for the unknown angle

We need to isolate $x$ in the equation

110 = 45 + x

Subtract $45$ from both sides:

110 - 45 = x

Compute the difference

Now calculate $110 - 45$ :

110 - 45 = 65

So, the measure of the other remote interior angle is $65^\circ$ , which corresponds to choice C.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

Hints

Desmos Guide

Step-by-step Explanation