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### Congruent Triangles in Parallelograms: Proof

```Date: 06/29/2003 at 14:40:22
From: Santan Paul

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

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

Good luck,

- Doctor Jaffee, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Triangles and Other Polygons

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