Congruent Triangles in Parallelograms: Proof
Date: 06/29/2003 at 14:40:22 From: Santan Paul Subject: About Parallelogram. O is a point inside triangle PQR. The parallelograms QORX, ROPY, and POQZ are drawn. Prove that triangle PQR is congruent to triangle XYZ.
Date: 06/29/2003 at 15:47:56 From: Doctor Jaffee Subject: Re: About Parallelogram. Hi Santan Paul, The hardest part of solving this problem for me was drawing the picture. I started by making a rough sketch and as a result became nearly convinced that the theorem was false. However, I decided to construct a Geometer's Sketchpad sketch to verify my conjecture and I discovered that I was wrong and that the theorem could be proved. Once I had a good picture, it was easy. So, get a large piece of paper and use the entire paper for your sketch. Draw triangle PQR. Place the point O anywhere inside it and construct parallellograms QORX, ROPY and POQZ. There are three angles at point O. Let angle POQ measure 'a' degrees, angle ROQ measure 'b' degrees, and angle POR measure 'c' degrees. It follows that a + b + c = 360. Since ROQX and ROPY are parallelograms, the measure of angle ORX must be 180 - b and the measure of angle ORY must be 180 - c. Put those two angles together and you have angle XRY whose measure is 360 - b - c. Since a + b + c = 360, 360 - b - c = a. Therefore, the measure of angle XRY is 'a'. The measure of angle PZQ is also 'a'. Furthermore, PO = YR and PO = ZQ since they are opposite sides of parallelograms. By transitivity, YR = ZQ. Likewise, PZ = RX. We can now justify that triangle PZQ is congruent to triangle XRY by SAS. Thus, PQ = XY since they are corresponding parts of the two congruent triangles. Likewise, you should be able to prove that YZ = RQ and that PR = ZX. Once you've done that, finishing the proof is easy. Give it a try and if you want to check your solution with me or if you have difficulties or other questions, write back to me and I'll try to help you some more. Good luck, - Doctor Jaffee, The Math Forum http://mathforum.org/dr.math/
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