Daily Regents: Parallelogram Proof (June 2015)

Even though the title says "Daily", I won't be doing them daily any more, until the exams get closer. But they will be frequent.

Geometry, June 2015, Question 33

33. Given: Quadrilateral ABCD is a parallelogram with diagonals AC and BD intersecting at E.
Prove: Triangle AED = Triangle CEB

You can write a two-column proof or a paragraph proof. I am going to use a paragraph proof to show that they are congruent using SSS. That means that I need to show three sets of sides are congruent. You can also show this using ASA or SAS, but remember to show all three pairs. Don't leave any out.

ABCD is given as a parallelogram. AD is congruent to BC because the opposite sides of a parallelogram are congruent. AC and BD are given as the diagonals of the parallelogram. AC and BD bisect each other because the diagonals of a parallelogram bisect each other. This make AE congruent to CE and BE congruent to DE because of the definition of bisect. Therefore, triangle AED is congruent to triangle CEB because of SSS.

If you wanted to use ASA, you could do so by pointing out that AC and BD are transversals across parallel lines. After that, use Alternate Interior Angles to show EAD = ECB and EDA = EBC.

For SAS, you need a combination of the steps above.

Also, you know that there are vertical angles at point E which are congruent.

Many approaches to a correct answer.