What you’ll learn
Prove parallelogram properties.
Key idea
Use both opposite-side parallel relations to obtain two angle pairs, then identify their included shared side.
Parallelism alone does not fix segment length; both angle pairs and shared included diagonal precede corresponding-side equality.
Worked example
Complete missing statements or reasons in a proof for 'Prove parallelogram properties'.
Parallelogram A=(0, 0), B=(2, 0), C=(3, 2), D=(1, 2); both AB parallel CD and BC parallel AD are given. Diagonal AC is drawn; proof establishes BAC=DCA and BCA=DAC, with AC=CA. Complete the triangle congruence/reason and AB/CD consequence.
- Draw AC. AB/CD parallel gives angle BAC=DCA; BC/AD parallel gives angle BCA=DAC by alternate interior angles.
- Included AC=CA is shared, so triangle BAC congruent to DCA by ASA. Corresponding sides AB/CD equal, here length 2; parallelism alone was not sufficient.
Answer: Triangle BAC congruent to DCA by ASA; corresponding AB=CD=2.
Quick Check
Complete missing statements or reasons in a proof for 'Prove parallelogram properties'.
Parallelogram A=(0, 0), B=(3, 0), C=(2, 2), D=(-1, 2); both AB parallel CD and BC parallel AD are given. Diagonal AC is drawn; proof establishes BAC=DCA and BCA=DAC, with AC=CA. Complete the triangle congruence/reason and AB/CD consequence.