A rombus that just doesn't want to show it's angles?

Question

http://www.rmckenna.org/files/026/954/quiz3.jpg

this is the link to it, I'm supposed to find LPQ but I can't seem to compare QLM or QPM

Answer 1

In a rhombus, the opposite angles are congruent and the adjacent angles are complementary. So we know that QLM = QPM, or

2x^2 - 10 = 8x

x^2 - 4x - 5 = 0

(x-5)(x+1) = 0

If x = -1, then QPM = -8 < 0, which is nonsensical (an angle should have a positve measure), so x = 5, QPM = QLM = 8x = 40.

Since LPQ + QPM = 180, we have that LPQ = 140