Look at the quadrilateral shown below: A quadrilateral ABCD is shown with diagonals AC and BD intersecting in point O. Angle AOB is labeled as 1, angle BOC is labeled as 4, angle COD is labeled as 2, and angle AOD is labeled as 3.

Terra writes the following proof for the theorem: If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram:

Terra's proof

AO = OC because it is given that diagonals bisect each other.
BO = OD because it is given that diagonals bisect each other.
For triangles AOB and COD, angle 1 is equal to angle 2, as they are .
Therefore, the triangles AOB and COD are congruent by SAS postulate.
Similarly, triangles AOD and COB are congruent.
By CPCTC, angle ABD is equal to angle BDC and angle ADB is equal to angle DBC.
As the alternate interior angles are congruent, the opposite sides of quadrilateral ABCD are parallel.
Therefore, ABCD is a parallelogram.