Clara writes the following proof for the theorem: if the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram: clara's proof 1. for triangles aob and cod, angle 1 is equal to angle 2 as they are vertical angles. 2. ao = oc and bo = od because it is given that diagonals bisect each other. 3. the triangles aob and cod are congruent by sas postulate. 4. similarly, triangles aod and cob are congruent. 5. by angle abd is equal to angle cdb and angle adb is equal to angle cbd. 6. as the alternate interior angles are congruent the opposite sides of quadrilateral abcd are parallel. 7. therefore, abcd is a parallelogram.