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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 
Associated Topics:
High School Triangles and Other Polygons

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