Ab = 2x + 20 ad = 3x + 23 bc = 3 - x cd = 4x + 30 quadrilateral abcd is a parallelogram if one pair of opposite sides is both equal and parallel. prove that quadrilateral abcd is a parallelogram by finding the value of x.

the value of x that makes ABCD be a parallelogram is x = -5

We can set the lengths of opposite sides equal to find a value for x that makes them so.

... AB = CD

... 2x +20 = 4x +30

... 0 = 2x +10 . . . . . . . . subtract the left side of the equation

... 0 = x +5 . . . . . . . . . . divide by 2

... -5 = x . . . . . . . . . . . . add -5

When x has the value -5, AB = CD. We can check the lengths of the other sides for that value of x:

.. BC = 3 -x = 3 -(-5) = 8

.. AD = 3x +23 = 3(-5) +23 = 8

Then for x = -5, we also have AD = BC

Thus for x = -5 opposite sides of ABCD are of equal length. The figure will be a parallelogram for that value of x.

Comment on the problem

The problem asks to prove the shape of ABCD by finding x. If we find x to be -6, the side lengths are 8, 5, 9, 6, and the figure is decidedly not a parallelogram. We can choose x (within bounds) to make the shape almost anything. It is not possible to prove the shape is a parallelogram, except for a specific value of x.

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