# 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.
• 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.
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. 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.
StatementsReasons
1. Parallelogram ABCD has diagonal ¯AC Given
2. ΔABC ~= ΔCDA Theorem 15.5