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Geometry and Trigonometry / Lines, angles, and triangles Difficulty: Medium

In the figure, line $q$ is parallel to line $r$ , and both lines are intersected by line $s$ . If $y = 2 x + 8$ , what is the value of $x$ ?

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Explanation

The correct answer is $57$ . Based on the figure, the angle with measure $y °$ and the angle vertical to the angle with measure $58 degree$ are same side interior angles. Since vertical angles are congruent, the angle vertical to the angle with measure $58 degree$ also has measure $58 degree$ . It’s given that lines $q$ and $r$ are parallel. Therefore, same side interior angles between lines $q$ and $r$ are supplementary. It follows that $y + 58 = 180$ . If $y = 2 x + 8$ , then the value of $x$ can be found by substituting $2 x plus 8$ for $y$ in the equation $y + 58 = 180$ , which yields $(2 x + 8) + 58 = 180$ , or $2 x + 66 = 180$ . Subtracting $66$ from both sides of this equation yields $2 x equals 114$ . Dividing both sides of this equation by $2$ yields $x equals 57$ . Thus, if $y = 2 x + 8$ , the value of $x$ is $57$ .