Converse Parallelogram Opposite Angles Theorem

Assume that ABCD is a quadrilateral with opposite congruent angles. It should be noted that congruent angles have the same measure. Then, let x^(∘) be the measure of ∠ A and ∠ C, and y^(∘) be the measure of ∠ B and ∠ D.

By the Polygon Interior Angles Theorem, the sum of the interior angles of a quadrilateral is 360^(∘). With this information, a relation can be found between the consecutive interior angles of ABCD.

Since x+y=180, the consecutive interior angles of ABCD are supplementary. Therefore, by the Converse Consecutive Interior Angles Theorem, it can be concluded that opposite sides of ABCD are parallel.

Consequently, by the definition of a parallelogram, ABCD is a parallelogram.

Completed Proof

2 &Given:&& ∠ A ≅ ∠ C and ∠ B ≅ ∠ D &Prove:&& ABCD is a parallelogram. Proof: