Question

m ∠ DAB = 85 − 2x m ∠ ABC = 90 + x m ∠ BCD = 80 − 3x m ∠ CDA = 105 + 4x Quadrilateral ABCD is a parallelogram if pairs of consecutive angles are supplementary. Prove that Quadrilateral ABCD is a parallelogram by finding the value of x.

Answer 1

Two angles are supplementary if they add up to 180 degrees.
 We have then:
 m ∠ DAB + m ∠ ABC = 180
 Substituting:
 (85 - 2x) + (90 + x) = 180
 From here, we clear the value of x:
 -x + 175 = 180
 x = 175 - 180 
 x = - 5
 Let's try now that it is a parallelogram.
 The opposite angles of a parallelogram are equal in measure:
 m ∠ DAB = m ∠ BCD
 85 - 2x = 80 - 3x
 85 - 2 (-5) = 80 - 3 (-5)
 85 + 10 = 80 +15
 95 = 95
 Answer:
 x = -5
 m ∠ DAB = m ∠ BCD (It is shown to be a parallelogram)