Two lines intersect, forming four angles. One of the angles measures , and the angle vertical to it measures . What is the value of ?

Solution available with step-by-step explanation

SAT Lines, Angles & Triangles Question 7 | Free SAT Prep | Aniko

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Question 7·Easy·Lines, Angles, and Triangles

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Two lines intersect, forming four angles. One of the angles measures $(5x + 4)^\circ$ , and the angle vertical to it measures $(7x - 20)^\circ$ . What is the value of $x$ ?

For line-and-angle questions, first identify the relationship: vertical angles are equal, linear pairs are supplementary, and angles around a point sum to $360^\circ$ . Translate that relationship into a simple equation (equal for congruent angles, sum to a known total for supplementary or around a point), then solve the equation carefully, keeping track of signs and combining like terms step by step. This approach is fast and avoids guessing from the choices.

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Identify the angle relationship

The two angles described are vertical angles. What is always true about vertical angles?

Translate the relationship into an equation

If two angles are vertical and therefore congruent, how can you relate $5x + 4$ and $7x - 20$ in an equation?

Solve carefully

After you set the two expressions equal, isolate $x$ step by step. Watch your signs when moving terms from one side of the equation to the other.

Desmos Guide

Enter both angle expressions as lines

In Desmos, type y = 5x + 4 on one line and y = 7x - 20 on another. These represent the measures of the two vertical angles as functions of $x$ .

Find the intersection point

Look for the point where the two lines intersect. Click on the intersection; the ordered pair $(x, y)$ will appear. The x-coordinate of this intersection is the value of $x$ that makes the two angle measures equal.

Step-by-step Explanation

Use the property of vertical angles

When two lines intersect, they form pairs of vertical angles (the ones directly across from each other). Vertical angles are always equal in measure.

So, if one angle measures $(5x + 4)^\circ$ and the angle vertical to it measures $(7x - 20)^\circ$ , their measures must be the same.

Set up the equation

Because the two vertical angles are equal, write an equation that sets their expressions equal:

5x + 4 = 7x - 20

Now you just need to solve this linear equation for $x$ .

Solve for x

Solve the equation step by step:

First, subtract $5x$ from both sides:

\begin{aligned} 5x + 4 &= 7x - 20 \\ 4 &= 2x - 20 \end{aligned}

Next, add $20$ to both sides:

\begin{aligned} 4 + 20 &= 2x \\ 24 &= 2x \end{aligned}

Finally, divide both sides by $2$ :

\begin{aligned} \frac{24}{2} &= x \\ 12 &= x \end{aligned}

So, the value of $x$ is $12$ , which corresponds to answer choice A.

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation