For a shape to be a parallelogram, the opposite interior angles must be equal and the adjacent interior angles must sum to 180 degrees. That being said, angle L and angle Q must be equal and angle P + angle Q = 180 degrees.
Given the following data:
m = 14 and n = 12.5
Angle L = 5m + 36 = 5(14) + 36 = 106
Angle Q = 4m + 50 = 4(14) + 50 = 106
Angle P = 6n - 1 = 6(12.5) - 1 = 74
Angle K = 360 - 106 - 106 - 74 = 74
From the given data above, we can see that the opposite interior angles Angle L and Angle Q are congruent with an angle of 106 degrees. In addition, we can also see that angle Q and Angle P are supplementary angles because 106 and 74 when added is equal to 180 degrees. Since the sum of the interior angles of a quadrilateral is 360 degrees, we can solve Angle K by subtracting 106, 106, and 74 from 360. Therefore, angle K is equal to 74 degrees. Since the other opposite interior angles, Angle P and Angle K, are also congruent to each other, then, we can say that KLPQ is a parallelogram.
The condition that proves it is "Opposite interior angles of a parallelogram are equal to each other."
Angle L is congruent to angle Q.
Angle P is congruent to Angle K.