What if you were given four pairs of coordinates that form a quadrilateral? How could you determine if that quadrilateral is a parallelogram? After completing this Concept, you'll be able to use the Parallel Congruent Sides Theorem and other quadrilateral theorems to solve problems like this one.
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CK-12 Proving a Quadrilateral is a Parallelogram
Guidance
Recall that a parallelogram is a quadrilateral with two pairs of parallel sides. Even if a quadrilateral is not marked with having two pairs of sides, it still might be a parallelogram. The following is a list of theorems that will help you decide if a quadrilateral is a parallelogram or not.
1) Opposite Sides Theorem Converse: If both pairs of opposite sides of a quadrilateral are congruent, then the figure is a parallelogram.
If then
2) Opposite Angles Theorem Converse: If both pairs of opposite angles of a quadrilateral are congruent, then the figure is a parallelogram.
If then
3) Parallelogram Diagonals Theorem Converse: If the diagonals of a quadrilateral bisect each other, then the figure is a parallelogram.
If then
4) Parallel Congruent Sides Theorem: If a quadrilateral has one set of parallel lines that are also congruent, then it is a parallelogram.
If then
You can use any of the above theorems to help show that a quadrilateral is a parallelogram. If you are working in the plane, you might need to know the formulas shown below to help you use the theorems.
- The Slope Formula, . (Remember that if slopes are the same then lines are parallel).
- The Distance Formula, . (This will help you to show that two sides are congruent).
- The Midpoint Formula, . (If the midpoints of the diagonals are the same then the diagonals bisect each other).
Example A
Prove the Opposite Sides Theorem Converse.
Given :
Prove : is a parallelogram
Statement | Reason |
---|---|
1. | 1.Given |
2. | 2. Reflexive PoC |
3. | 3. SSS |
4. | 4. CPCTC |
5. | 5. Alternate Interior Angles Converse |
6. is a parallelogram | 6. Definition of a parallelogram |
Example B
Is quadrilateral a parallelogram? How do you know?
a) By the Opposite Angles Theorem Converse, is a parallelogram.
b) is not a parallelogram because the diagonals do not bisect each other.
Example C
Is the quadrilateral a parallelogram?
Let’s use the Parallel Congruent Sides Theorem to see if is a parallelogram. First, find the length of and using the distance formula.
Next find the slopes to check if the lines are parallel.
and the slopes are the same (implying that the lines are parallel), so is a parallelogram.
CK-12 Proving a Quadrilateral is a Parallelogram
Guided Practice
1. Prove the Parallel Congruent Sides Theorem.
Given : , and
Prove : is a parallelogram
2. What value of would make a parallelogram?
3. Is the quadrilateral a parallelogram?
Answers:
1.
Statement | Reason |
---|---|
1. , and | 1. Given |
2. | 2. Alternate Interior Angles |
3. | 3. Reflexive PoC |
4. | 4. SAS |
5. | 5. CPCTC |
6. is a parallelogram | 6. Opposite Sides Converse |
2. . By the Parallel Congruent Sides Theorem, would be a parallelogram if .
3. Let’s use the Parallelogram Diagonals Converse to see if is a parallelogram. Find the midpoint of each diagonal.
is not a parallelogram because the midpoints are not the same.
Practice
For questions 1-12, determine if the quadrilaterals are parallelograms.
For questions 13-18, determine the value of and that would make the quadrilateral a parallelogram.
For questions 19-22, determine if is a parallelogram.
Fill in the blanks in the proofs below.
- Opposite Angles Theorem Converse
Given :
Prove : is a parallelogram
Statement | Reason |
---|---|
1. | 1. |
2. | 2. |
3. | 3. Definition of a quadrilateral |
4. | 4. |
5. | 5. Combine Like Terms |
6. | 6. Division PoE |
7. and are supplementary and are supplementary | 7. |
8. | 8. Consecutive Interior Angles Converse |
9. is a parallelogram | 9. |
- Parallelogram Diagonals Theorem Converse
Given :
Prove : is a parallelogram
Statement | Reason |
---|---|
1. | 1. |
2. | 2. Vertical Angles Theorem |
3. | 3. |
4. | 4. |
5. is a parallelogram | 5. |
- Given : Prove : is a parallelogram
Statement | Reason |
---|---|
1. | 1. |
2. | 2. |
3. is a parallelogram | 3. |