In , the measure of is twice the measure of , and the measure of is . What is the measure of ?

Solution available with step-by-step explanation

SAT Lines, Angles & Triangles Question 43 | Free SAT Prep | Aniko

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Question 43·Easy·Lines, Angles, and Triangles

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In $\triangle PQR$ , the measure of $\angle P$ is twice the measure of $\angle Q$ , and the measure of $\angle R$ is $30^\circ$ . What is the measure of $\angle P$ ?

For triangle angle questions where one angle is described in terms of another, quickly assign a variable to one unknown angle and express the others in terms of that variable. Use the fact that the three interior angles add to $180^\circ$ to write a single equation, solve for the variable, and then carefully answer what the question actually asks for (here, angle P, not the variable itself). This keeps your work organized and avoids guessing based on special triangles like right or equilateral triangles.

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Recall a key triangle fact

What is the total sum of the measures of the interior angles of any triangle?

Introduce a variable

Try letting $\angle Q$ be $x^\circ$ . How can you write $\angle P$ in terms of $x$ using the fact that $\angle P$ is twice $\angle Q$ ?

Set up and solve the equation

Use your expressions for $\angle P$ , $\angle Q$ , and the given $30^\circ$ for $\angle R$ in the triangle sum equation. Solve for $x$ , then use it to find the measure of $\angle P$ .

Desmos Guide

Graph the equation for the triangle's angles

In Desmos, enter the first equation as y = 3x + 30 (this represents $\angle P + \angle Q + \angle R$ in terms of $x$ , with $\angle Q = x$ and $\angle P = 2x$ ).

Graph the total angle sum

Enter a second equation y = 180 to represent the total sum of the angles in a triangle.

Find angle Q from the intersection

Use the intersection tool (tap where the two graphs cross) to find the $x$ -value where 3x + 30 equals 180. That $x$ -value is the measure of $\angle Q$ in degrees.

Compute angle P from angle Q

In an expression line, type 2 * (value_of_x) using the $x$ -value you found (for example, 2 * 50 if $x=50$ ). The result shown is the measure of $\angle P$ in degrees.

Step-by-step Explanation

Use the triangle angle sum fact

In any triangle, the measures of the three interior angles always add up to $180^\circ$ . So for $\triangle PQR$ :

\angle P + \angle Q + \angle R = 180^\circ

Represent the angles with a variable

Let the measure of $\angle Q$ be $x^\circ$ .

Then, because $\angle P$ is twice $\angle Q$ , we have $\angle P = 2x^\circ$ .
We are given that $\angle R = 30^\circ$ .

Substitute these into the triangle sum equation:

2x + x + 30 = 180

Solve for the measure of angle Q

Combine like terms on the left side:

3x + 30 = 180

Subtract 30 from both sides:

3x = 150

Divide both sides by 3:

x = 50

So $\angle Q = 50^\circ$ .

Find the measure of angle P and choose the answer

Since $\angle P$ is twice $\angle Q$ and $\angle Q = 50^\circ$ :

\angle P = 2x = 2 \cdot 50 = 100

So the measure of $\angle P$ is $100^\circ$ , which corresponds to answer choice C.

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Step-by-step Explanation

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