Answer:
the value of x that makes ABCD be a parallelogram is x = -5
Step-by-step explanation:
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
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Thus for x = -5 opposite sides of ABCD are of equal length. The figure will be a parallelogram for that value of x.
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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.