In the mentioned exercise, we were given a diagram of a quadrilateral and were asked what value of x made the quadrilateral a parallelogram.
We started solving the problem by noting that, according to the Parallelogram Diagonals Converse, if the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram. Therefore, for the given quadrilateral to be a parallelogram, both parts of each diagonal must have the same length.
Using this fact, we started by equating the lengths of the two parts of both diagonals.
2x+1 = x+6 and 4x-2 = 3x+3
We then solved both equations and found that the quadrilateral is a parallelogram when x is equal to 5.
Alternative Start
Alternatively, we could have used the Parallelogram Opposite Sides Converse. This tells us that if the opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. For the opposite sides of a quadrilateral to be congruent, they must have equal lengths.
Using this fact, we could have written a different set of two equations for x.
ccc
5x = 3x+10 & and & 4x-4 = 3x+1
We can then solve both equations and find a single value of x that satisfies both of them. This value is also 5.
