Lines, angles, and triangles SAT question

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 $x^{\circ }$ and the angle with measure $y^{\circ }$ are vertical angles, so $x=y$. It's given that $x=6k+13$ and $y=8k-29$. Substituting $6k+13$ for $x$ and $8k-29$ for $y$ in the equation $x=y$ yields $6k+13=8k-29$. Subtracting $6k$ from both sides of this equation yields $13=2k-29$. Adding $29$ to both sides of this equation yields $42=2k$, or $2k=42$. Dividing both sides of this equation by $2$ yields $k=21$. 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 $y^{\circ }$ and $z^{\circ }$ are supplementary, so $y+z=180$. Substituting $8k-29$ for $y$ in this equation yields $8k-29+z=180$. 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 $z=41$. 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 $z$.