Answer:
13. (x, y) = (21, 15)
15. (x, y) = (16, 23)
16. (x, y) = (9, 13)
Explanation:
You want the values of x and y where angles at a transversal are expressed in terms of these variables.
Where a transversal crosses parallel lines, all acute angles are congruent, and all obtuse angles are congruent. The acute and obtuse angles are supplementary.
Vertical angles are always congruent. Angles of a linear pair are supplementary. These relations are used to write the necessary systems of equations for finding x and y.
13.
Consecutive interior angles are supplementary:
(3x -16) +(6x +7) = 180
9x -9 = 180 . . . . . simplify
x -1 = 20 . . . . . . . divide by 9 (because we can)
x = 21
Vertical angles are congruent:
(6x +7) = (11y -32)
6(21) +7 +32 = 11y . . . . . add 32, substitute for x
y = 165/11 = 15 . . . . . . divide by 11
(x, y) = (21, 15)
15.
Alternate exterior angles are congruent:
4x +4 = 7x -44
48 = 3x . . . . . . . . add 44-4x
16 = x . . . . . . divide by 3
Consecutive exterior angles are supplementary:
39 + (8y -43) = 180
8y = 184 . . . . . . . . . . . . add 4
y = 23 . . . . . . . . . . . . divide by 8
(x, y) = (16, 23)
16.
Alternate exterior angles are congruent:
(15x -26) = (12x +1)
3x = 27 . . . . . . . . . . . . add 26-12x
x = 9 . . . . . . . . . . . divide by 3
Angles of a triangle total 180°. Vertical angles are congruent.
28 +(12x +1) +(4y -9) = 180
12x +4y = 160 . . . . . . . . . . subtract 20
3x +y = 40 . . . . . . . . . divide by 4 (because we can)
y = 40 -3x = 40 -3(9) = 13
(x, y) = (9, 13)
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Additional comments
We end up with "3-step" equations that have variables and constants on both sides. In each case, we subtract the variable term with the lowest coefficient, and the constant from the other side. We subtract both these in one step, rather than the usual two steps. It saves a bit of writing and gets you to the same place: variable term on one side, and constant on the other side of the equal sign.
The "because we can" steps factor out a common factor from all of the terms of the equation. This lets us deal with smaller numbers, and generally leaves the variable with a coefficient of 1. The total number of steps does not change, just the order. We could subtract, then divide, and we would end up in the same place.
The relations between angles where lines cross are important to setting up the equations. It is helpful to learn them. (The actual relations are not complicated. There are many names and definitions used to describe them, so there's a lot of vocabulary around basically simple relations.)