Parallelogram Consecutive Angles Theorem

The Parallelogram Consecutive Angles Theorem states that the consecutive angles of a parallelogram are supplementary to each other. To prove this theorem take the generic parallelogram ABCD.

Since the figure is a parallelogram

AB ǀǀ CD

AD acts as the transversal cutting through the parallel lines AB and CD. Because of the Consecutive Interior Angles Theorem.

∠A + ∠D = 180º                                                           1

The definition of supplementary angles states:

∴ ∠A and ∠D are supplementary

Using the sum of the interior angles of a quadrilateral:

∠A + ∠B + ∠C + ∠D = 360º                                            2

Subbing equation 1 into equation 2 gives us:

180º + ∠B + ∠C = 360º

The subtraction equivalence property

∠B + ∠C = 180º

Definition of supplementary angles

∴ ∠B and ∠C are supplementary angles