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

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SAT Lines, Angles & Triangles Question 4 | Free SAT Prep | Aniko

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Question 4·Easy·Lines, Angles, and Triangles

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In $\triangle LMN$ , $\angle L$ is twice the measure of $\angle M$ , and $\angle N$ is $60^\circ$ greater than $\angle M$ . What is the measure of $\angle L$ ?

For triangle angle problems with relationships in words (like “twice” or “60 degrees greater”), first assign a variable to the simplest angle, usually the one mentioned in all relationships. Express the other angles in terms of that variable, then use the fact that the three interior angles sum to $180^\circ$ to create an equation. Solve for the variable carefully, and finally check what the question is actually asking for—often it wants one of the other angles, not the variable you originally defined.

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Use the triangle angle sum

Recall that the sum of the measures of the three interior angles in any triangle is $180^\circ$ . How can you use this fact here?

Choose a variable for one angle

Let angle $M$ be $x$ . Then express angle $L$ and angle $N$ in terms of $x$ using the relationships given in the problem.

Write an equation and solve for the variable

Add your expressions for angles $L$ , $M$ , and $N$ , set the sum equal to $180$ , and solve the resulting equation for $x$ .

Answer what is actually being asked

After you find $x$ (which represents angle $M$ ), remember the question is asking for the measure of angle $L$ . Use the relationship between angle $L$ and angle $M$ to get the final value.

Desmos Guide

Model the equation in Desmos

In Desmos, enter the expression y = 4x + 60 (this represents the sum of the three angle expressions). Then enter y = 180 as a second line (this represents the total degrees in a triangle).

Find the value of angle M

Look at the intersection point of the two lines. The $x$ -coordinate of this point is the value of $x$ , which is the measure of angle $M$ .

Use Desmos result to get angle L

Once you know $x$ , compute $2x$ (you can type 2*(value) in Desmos) to find the measure of angle $L$ , since angle $L$ is twice angle $M$ .

Step-by-step Explanation

Represent the angles with a variable

Let angle $M$ be $x$ degrees.

Angle $L$ is twice angle $M$ , so angle $L = 2x$ .
Angle $N$ is $60^\circ$ greater than angle $M$ , so angle $N = x + 60$ .

Use the triangle angle sum

In any triangle, the three interior angles add up to $180^\circ$ .

So for $\triangle LMN$ :

\text{angle }L + \text{angle }M + \text{angle }N = 180

Substitute the expressions for each angle in terms of $x$ :

2x + x + (x + 60) = 180

Solve for angle M

First simplify the equation:

\begin{aligned} 2x + x + (x + 60) &= 180 \\ 4x + 60 &= 180 \end{aligned}

Now solve for $x$ :

\begin{aligned} 4x + 60 &= 180 \\ 4x &= 180 - 60 = 120 \\ x &= 30 \end{aligned}

So angle $M$ measures $30^\circ$ .

Find angle L and answer the question

The problem asks for the measure of angle $L$ , which is $2x$ .

We found $x = 30$ , so:

\text{angle }L = 2x = 2 \cdot 30 = 60^\circ

So the measure of angle $L$ is $60^\circ$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation