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.

• 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.

• Given: Parallelogram ABCD with diagonal ¯AC.
• Prove: ΔABC ~= ΔCDA.
• 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 ΔABC with two corresponding angles in ΔCDA. Because ¯AC ~= ¯AC, you can use the ASA Postulate to show ΔABC ~= ΔCDA.

	Statements	Reasons
1.	Parallelogram ABCD has diagonal ¯AC	Given
2.	¯BC ‌ ‌ ¯AD cut by transversal ¯AC	Definition of transversal
3.	∠BAC and ∠DCA are alternate interior angles	Definition 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 angles	Definition of alternate interior angles
7.	∠ACB ~= ∠DAC	Theorem 10.2
8.	¯AC ~= ¯AC	Reflexive property of ~=
9.	ΔABC ~= ΔCDA	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.

	Statements	Reasons
1.	Parallelogram ABCD has diagonal ¯AC	Given
2.	ΔABC ~= ΔCDA	Theorem 15.5
3.	¯BC ~= ¯AD	CPOCTAC

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 ΔAMD and ΔCMB. 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 ΔAMD ~= ΔCMB. Finish it up by using CPOCTAC.