6.2: Properties of Parallelograms
Learning Objectives
- Define a parallelogram.
- Understand the properties of a parallelogram
- Apply theorems about a parallelogram’s sides, angles and diagonals.
Review Queue
- Draw a quadrilateral with one set of parallel sides.
- Draw a quadrilateral with two sets of parallel sides.
- Find the measure of the missing angles in the quadrilaterals below.
Know What? A college has a parallelogram-shaped courtyard between two buildings. The school wants to build two walkways on the diagonals of the parallelogram and a fountain where they intersect. The walkways are going to be 50 feet and 68 feet long. Where would the fountain be?
What is a Parallelogram?
Parallelogram: A quadrilateral with two pairs of parallel sides.
Notice that each pair of sides is marked parallel. Also, recall that two lines are parallel when they are perpendicular to the same line. Parallelograms have a lot of interesting properties.
Investigation 6-2: Properties of Parallelograms
Tools Needed: Paper, pencil, ruler, protractor
- Draw a set of parallel lines by drawing a 3 inch line on either side of your ruler.
- Rotate the ruler and repeat so you have a parallelogram. If you have colored pencils, outline the parallelogram in another color.
- Measure the four interior angles of the parallelogram as well as the length of each side. What do you notice?
- Draw the diagonals. Measure each and then measure the lengths from the point of intersection to each vertex.
To continue to explore the properties of a parallelogram, see the website: http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/interactive-parallelogram.php
Opposite Sides Theorem: If a quadrilateral is a parallelogram, then the opposite sides are congruent.
If then
Opposite Angles Theorem: If a quadrilateral is a parallelogram, then the opposite angles are congruent.
If then
Consecutive Angles Theorem: If a quadrilateral is a parallelogram, then the consecutive angles are supplementary.
If then
Parallelogram Diagonals Theorem: If a quadrilateral is a parallelogram, then the diagonals bisect each other.
If then
Proof of Opposite Sides Theorem
Given: is a parallelogram with diagonal
Prove:
Statement | Reason |
---|---|
1. is a parallelogram with diagonal | Given |
2. | Definition of a parallelogram |
3. | Alternate Interior Angles Theorem |
4. | Reflexive PoC |
5. | ASA |
6. | CPCTC |
The proof of the Opposite Angles Theorem is almost identical. For the last step, the angles are congruent by CPCTC.
Example 1: is a parallelogram. If , find the measure of the other angles.
Solution: Draw a picture. When labeling the vertices, the letters are listed, in order, clockwise.
If , then by the Opposite Angles Theorem.
Example 2: Algebra Connection Find the values of and .
Solution: Opposite sides are congruent.
Diagonals in a Parallelogram
From the Parallelogram Diagonals Theorem, we know that the diagonals of a parallelogram bisect each other.
Example 3: Show that the diagonals of bisect each other.
Solution: Find the midpoint of each diagonal.
Because they are the same point, the diagonals intersect at each other’s midpoint. This means they bisect each other.
This is one way to show a quadrilateral is a parallelogram.
Example 4: Algebra Connection is a parallelogram and and . Solve for .
Solution:
Know What? Revisited The diagonals bisect each other, so the fountain is going to be 34 feet from either endpoint on the 68 foot diagonal and 25 feet from either endpoint on the 50 foot diagonal.
Review Questions
- Questions 1-6 are similar to Examples 2 and 4.
- Questions 7-10 are similar to Example 1.
- Questions 11-23 are similar to Examples 2 and 4.
- Questions 24-27 are similar to Example 3.
- Questions 28 and 29 are similar to the proof of the Opposite Sides Theorem.
- Question 30 is a challenge. Use the properties of parallelograms.
is a parallelogram. Fill in the blanks below.
- If , then ______.
- If , then ______.
- If = ______.
- If = ______.
- If = ______.
- If , then = ______.
- If in parallelogram , find the other three angles.
- If in parallelogram , find the other three angles.
- If in parallelogram , find the measure of all four angles.
- If in parallelogram , find the other three angles.
For questions 11-19, find the measures of the variable(s). All the figures below are parallelograms.
Use the parallelogram to find:
Find the point of intersection of the diagonals to see if is a parallelogram.
Fill in the blanks in the proofs below.
- Opposite Angles Theorem
Given: is a parallelogram with diagonal
Prove:
Statement | Reason |
---|---|
1. | Given |
2. | |
3. | Alternate Interior Angles Theorem |
4. | Reflexive PoC |
5. | |
6. |
- Parallelogram Diagonals Theorem
Given: is a parallelogram with diagonals and
Prove:
Statement | Reason |
---|---|
1. | |
2. | Definition of a parallelogram |
3. | Alternate Interior Angles Theorem |
4. | |
5. | |
6. |
- Challenge Find and . (The two quadrilaterals with the same side are parallelograms.)