A B E D C AE = 3x - 6 , DE = 6y - 12 , EC = 2x + 2 and EB = 2y + 24 find the values of x and y for which ABCD must be a parallelogram. The drawing is not to scale. (4 points) x =; y =
To show that ABCD must be a parallelogram, we can use the fact that opposite sides of a parallelogram are congruent.
Given that AE = 3x - 6 and DE = 6y - 12, we can set them equal to each other:
3x - 6 = 6y - 12.
Solving for x, we get:
3x = 6y - 6.
x = 2y - 2. ----(1)
Given that EC = 2x + 2 and EB = 2y + 24, we can set them equal to each other:
2x + 2 = 2y + 24.
Solving for x, we get:
2x = 2y + 22.
x = y + 11. ----(2)
Now, we have two equations (1) and (2) for x in terms of y.
Substituting equation (2) into equation (1), we can solve for y:
2y - 2 = y + 11.
Simplifying this equation, we get:
y = 13.
Substituting y = 13 into equation (2), we can solve for x:
x = 13 + 11.
x = 24.
Therefore, the values of x and y for which ABCD must be a parallelogram are:
x = 24, y = 13.