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

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 xo and the angle with measure yo are vertical angles, so x = y. It's given that x = 6k + 13 and y = 8k - 29. Substituting 6k + 13 k 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 in angles with measures yo and zo 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.