Question 29·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

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.)

Question 29·Medium·Lines, Angles, and Triangles

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