Prove that if in the quadrilateral abcd angle A + angle B = angle B + angle C = 180 degrees, then ABCD is a parallelogram.

In order to prove that ABCD is a parallelogram, it is enough to prove that opposite lines in it are parallel. That is, AB ∥ SD; and BC ∥ AD. (1) For the proof, we use one of the properties of parallelism of lines:

Property: if two straight lines AB and CD at the intersection of the secant line BC form one-sided angles so that (<B + <C) = 180, then these straight lines are parallel. AB ∥ CD. Proven.

Similarly, at the intersection of two straight lines BC and AD of the third straight line AB, angles are formed so that <A + <B = 180, then straight BC ∥ AD.

We have proved property (1), so ABCD is a parallelogram.