Answer:
Explanation:
You want to know the values of x and y given expressions for three of the angles in a quadrilateral (trapezoid) cut by a diagonal, and the measure of a fourth angle.
Angle relations
There are some different angle relations that can be used here:
- alternate interior angles are congruent
- consecutive interior angles are supplementary
- the sum of angles in a triangle is 180°
The angles marked (2x+y)° and (5x-y)° are "alternate interior" angles, hence congruent:
2x +y = 5x -y
The angles marked (5x+y)° and (5x-y)° together comprise the lower left corner angle of the quadrilateral, which is supplementary to the angle marked 100°:
(5x +y) +(5x -y) = 180 -100
Solution
The first equation can be simplified to ...
2y = 3x . . . . . . add y-2x to both sides
The second equation can be simplified to ...
10x = 80
x = 8 . . . . . . . divide by 10
Then the first of these equations tells you the value of y:
2y = 3(8)
y = 12 . . . . . . divide by 2
The values of x and y are 8 and 12, respectively.
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Additional comment
There is nothing in the figure to suggest that the left and right sides of the quadrilateral are parallel. Hence the figure cannot be presumed to be a parallelogram. The one pair of parallel sides tells you it is a trapezoid.
For this problem, it only matters that the left side and the diagonal are transversals relative to the two parallel lines.
Clockwise in the upper triangle, the angle measures are 100°, 28°, 52°. The marked angle in the lower triangle is also 28°.