Answer:
- x = 6
- y = 17
- 18x +2y = 142 by the corresponding angles theorem, and
- 3x +2y = 52 by the alternate interior angles theorem
Step-by-step explanation:
Given a set of parallel lines with a transversal and angles marked, you want to find the values for x and y consistent with the markings.
Setup
Alternate interior angles are congruent, so the marked acute angles have the same measure:
 3x +2y = 52
Corresponding angles are congruent, so the marked obtuse angles have the same measure:
 18x +2y = 142
Solution
Subtracting the first equation from the second gives ...
 (18x +2y) -(3x +2y) = (142) -(52)
 15x = 90
 x = 6
The value of y can be found from either equation. We choose to use the first:
 3(6) +2y = 52
 9 +y = 26 . . . . . . . . divide by 2
 y = 17 . . . . . . . . . subtract 9
Summary
- x = 6
- y = 17
- 18x +2y = 142 by the corresponding angles theorem, and
- 3x +2y = 52 by the alternate interior angles theorem
