Question 162·Medium·Geometry and Trigonometry

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In the diagram, lines $m$ and $n$ represent two parallel roads. A cross street $t$ intersects both roads as shown. Two angles are labeled $(4x+12)^\circ$ and $(2x+6)^\circ$ .

Which choice is the value of $x$ ?

When a transversal crosses two parallel lines, first decide whether the marked angles are equal (corresponding/alternate interior) or add to $180^\circ$ (same-side interior/linear pair). Translate that relationship into one linear equation in $x$ , then solve and check that the resulting angle measures make sense.

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Identify the angle pair

The two labeled angles lie between the parallel lines $m$ and $n$ and are on the same side of transversal $t$ .

Write an equation using a sum

Same-side interior angles formed by a transversal with parallel lines add to $180^\circ$ .

Solve the linear equation

After you set up the equation, combine like terms and isolate $x$ .

Desmos Guide

Enter the supplementary equation

In Desmos, type: (4x+12) + (2x+6) = 180.

Solve for $x$

Use Desmos’s solution for $x$ (or type solve((4x+12)+(2x+6)=180,x) if available) and match it to the answer choices.

Step-by-step Explanation

Use the parallel-lines angle relationship

Since $m\parallel n$ and $t$ is a transversal, the labeled angles are same-side interior angles, so they are supplementary:

(4x+12)^\circ+(2x+6)^\circ=180^\circ.

Solve for $x$

Combine like terms and solve:

6x+18=180

6x=162

x=27.

So the correct choice is $27$ .

Question 162·Medium·Geometry and Trigonometry

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