Clockwise from top left, the 3 lines are labeled t, m, and n. Line t intersects both line m and

Question

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

Accepted Answer

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 /mo> and the angle with measure y /mo> are vertical angles, so x = y . It's given that x = 6 k + 13 and y = 8 k - 29 . Substituting 6 k + 13 for x and 8 k - 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 = 2 k , or 2 k = 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 /mo> and z /mo> are supplementary, so y + z = 180 . Substituting 8 k - 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 = 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 .

Question 551e10