In , the measures of and are and , respectively. The measure of is . What is the value of ? (Disregard the degree symbol when entering your answer.)

Solution available with step-by-step explanation

SAT Lines, Angles & Triangles Question 5 | Free SAT Prep | Aniko

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Question 5·Medium·Lines, Angles, and Triangles

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In $\triangle GHI$ , the measures of $\angle G$ and $\angle H$ are $k^\circ$ and $(k+40)^\circ$ , respectively. The measure of $\angle I$ is $50^\circ$ . What is the value of $k$ ? (Disregard the degree symbol when entering your answer.)

Your answer

(Express the answer as an integer)

For triangle angle problems, immediately use the fact that the three interior angles sum to 180°. Translate the angle descriptions into algebraic expressions, write a single equation that sets their sum equal to 180, and then solve the resulting one-variable linear equation carefully by combining like terms and isolating the variable. This direct setup avoids guessing and is very fast on the SAT.

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Recall a basic triangle fact

What is the sum of the measures of the three interior angles in any triangle?

Write an equation for the angle measures

Write an equation that adds $k$ , $(k+40)$ , and $50$ and sets that sum equal to the total angle sum of a triangle.

Solve the equation step by step

After you combine like terms, you should get a simple linear equation in $k$ . Isolate $k$ by undoing addition/subtraction first, then division.

Desmos Guide

Enter the angle-sum expressions as graphs

In Desmos, in the first line, type y = x + (x + 40) + 50. In the second line, type y = 180. Here, x is playing the role of $k$ .

Find the intersection point

Look for the intersection of the two graphs. Click on the intersection point; the x-coordinate shown there is the value of $k$ that makes the angle sum equal 180.

Step-by-step Explanation

Use the triangle angle-sum rule

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

So for $\triangle GHI$ :

$\angle G = k^\circ$
$\angle H = (k+40)^\circ$
$\angle I = 50^\circ$

These must satisfy:

k + (k+40) + 50 = 180

Simplify the equation

Combine like terms on the left side.

First add the $k$ terms and the constant terms:

k + (k+40) + 50 = 2k + 90

So the equation becomes

2k + 90 = 180

Solve for k

Solve the equation step by step.

Subtract 90 from both sides:

\begin{aligned} 2k + 90 - 90 &= 180 - 90 \\ 2k &= 90 \end{aligned}

Now divide both sides by 2:

\begin{aligned} \frac{2k}{2} &= \frac{90}{2} \\ k &= 45 \end{aligned}

So the value of $k$ is $45$ . (You would enter 45 as your answer.)

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