 Cite

# Geometry: Properties of Parallelograms

## Properties of Parallelograms

A parallelogram is a quadrilateral that has both pairs of opposite sides parallel. Parallelograms have many properties that are easy to prove using the properties of parallel lines. You will occasionally use a diagonal to divide a parallelogram into triangles. If you do this carefully, your triangles will be congruent, so you can use CPOCTAC.

##### Solid Facts

A parallelogram is a quadrilateral that has both pairs of opposite sides parallel.

• Theorem 15.5: A diagonal of a parallelogram separates it into two congruent triangles.
• Example 2: Write a formal proof of Theorem 15.5.
• Solution: Begin by going down the list of what you need to bring to a formal proof. We already have the statement of the theorem. Figure 15.7 shows parallelogram ABCD with diagonal AC. Figure 15.7Parallelogram ABCD with diagonal AC.

• Given: Parallelogram ABCD with diagonal AC.
• Prove: ?ABC ~= ?CDA.
• Proof: Your game plan is to make use of the properties of parallel lines cut by a transversal to relate two of the angles of ?ABC with two corresponding angles in ?CDA. Because AC ~= AC, you can use the ASA Postulate to show ?ABC ~= ?CDA.
StatementsReasons
1. Parallelogram ABCD has diagonal AC Given
2. BC ? ? AD cut by transversal AC Definition of transversal
3. ?BAC and ?DCA are alternate interior anglesDefinition of alternate interior angles
4. ?BAC ~= ?DCA Theorem 10.2
5. BC ? ? AD cut by transversal AC Definition of transversal
6. ?ACB and ?DAC are alternate interior anglesDefinition of alternate interior angles
7. ?ACB ~= ?DAC Theorem 10.2
8. AC ~= AC Reflexive property of ~=
9. ?ABC ~= ?CDA ASA Postulate

This theorem will come in handy when establishing theorems about parallelograms. A common technique involves using a diagonal to divide a parallelogram into two triangles and then applying CPOCTAC. The next two theorems use this technique. I'll prove the first one and let you prove the second.

• Theorem 15.6: Opposite sides of a parallelogram are congruent.
• Theorem 15.7: Opposite angles of a parallelogram are congruent.
• Example 3: Write a two-column proof of Theorem 15.6.
• Solution: You can draw from the information shown in Figure 15.7. We'll be dealing with the parallelogram ABCD and its diagonal AC. You will want to prove BC ~= AD.
StatementsReasons
1. Parallelogram ABCD has diagonal AC Given
2. ?ABC ~= ?CDA Theorem 15.5

The last property of a parallelogram that I will mention involves the intersection of the diagonals. It turns out that the diagonals of a parallelogram bisect each other. The proof of this is fairly straightforward, so I'll walk you through the game plan and let you provide the details.

• Theorem 15.8: The diagonals of a parallelogram bisect each other.

Take a look at parallelogram ABCD in Figure 15.8. It has diagonals AC and BD which intersect at M. We want to show AM ~= MC. The easiest way to do this is to find two triangles that are congruent and use CPOCTAC. The two triangles that we'll try to prove congruent are ?AMD and ?CMB. Because opposite sides of a parallelogram are congruent, BC ~= AD. Because vertical angles are congruent, ?AMD ~= CMB. Finally, we have BC ? ? AD cut by a transversal AC, and because ?BCA and ?CAD are alternate interior angles, they are congruent. Using the AAS Theorem, we can conclude that ?AMD ~= ?CMB. Finish it up by using CPOCTAC. Figure 15.8Parallelogram ABCD has diagonals AC and BD which intersect at M.

Excerpted from The Complete Idiot's Guide to Geometry 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

To order this book direct from the publisher, visit the Penguin USA website or call 1-800-253-6476. You can also purchase this book at Amazon.com and Barnes & Noble.