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Geometry and Trigonometry Difficulty: Medium

In the figure, lines $m$ and $n$ are parallel. If $x = 6 k + 13$ and $y = 8 k - 29$ , what is the value of $z$ ?

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Explanation

Choice C is correct. Vertical angles, which are angles that are opposite each other when two lines intersect, are congruent. The figure shows that lines $t$ and $m$ intersect. It follows that the angle with measure $x °$ and the angle with measure $y °$ are vertical angles, so $x = y$ . It's given that $x = 6 k + 13$ and $y = 8 k - 29$ . Substituting $6 k plus 13$ for $x$ and $8 k minus 29$ for $y$ in the equation $x = y$ yields $6 k + 13 = 8 k - 29$ . Subtracting $6 k$ from both sides of this equation yields $13 = 2 k - 29$ . Adding $29$ to both sides of this equation yields $42 equals 2 k$ , or $2 k equals 42$ . Dividing both sides of this equation by $2$ yields $k equals 21$ . It's given that lines $m$ and $n$ are parallel, and the figure shows that lines $m$ and $n$ are intersected by a transversal, line $t$ . If two parallel lines are intersected by a transversal, then the same-side interior angles are supplementary. It follows that the same-side interior angles with measures $y °$ and $z °$ are supplementary, so $y + z = 180$ . Substituting $8 k minus 29$ for $y$ in this equation yields $8 k - 29 + z = 180$ . Substituting $21$ for $k$ in this equation yields $8 (21) - 29 + z = 180$ , or $139 + z = 180$ . Subtracting $139$ from both sides of this equation yields $z equals 41$ . Therefore, the value of $z$ is $41$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the value of $k$ , not $z$ .

Choice D is incorrect. This is the value of $x$ or $y$ , not $z$ .