If 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.
To determine when ABCD must be a parallelogram, we need to find the conditions under which opposite sides are equal.
In a parallelogram, opposite sides are congruent. Therefore, we must have AE = CD and DE = BC.
Considering AE = CD, we have:
3x - 6 = 6y - 12
Simplifying this equation:
3x = 6y - 6
Dividing both sides by 3:
x = 2y - 2
Considering DE = BC, we have:
6y - 12 = 2y + 24
Simplifying this equation:
4y = 36
Dividing both sides by 4:
y = 9
Given y = 9, we can substitute this value back into the equation x = 2y - 2:
x = 2(9) - 2
x = 18 - 2
x = 16
Therefore, when x = 16 and y = 9, ABCD must be a parallelogram.