What if you were told that is a parallelogram and you are given the length of and the measure of ? What can you determine about , , , and ? After completing this Concept, you'll be able to apply parallelogram theorems to answer such questions.
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Guidance
A parallelogram is a quadrilateral with two pairs of parallel sides.
Notice that each pair of sides is marked parallel (for the last two shapes, remember that when two lines are perpendicular to the same line then they are parallel). Parallelograms have a lot of interesting properties. (You can explore the properties of a parallelogram at: http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/interactive-parallelogram.php )
Facts about Parallelograms
1) Opposite Sides Theorem: If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent.
If then
2) Opposite Angles Theorem: If a quadrilateral is a parallelogram, then both pairs of opposite angles are congruent.
If then
3) Consecutive Angles Theorem: If a quadrilateral is a parallelogram, then all pairs of consecutive angles are supplementary.
If then
4) Parallelogram Diagonals Theorem: If a quadrilateral is a parallelogram, then the diagonals bisect each other.
If then
Example A
is a parallelogram. If , find the measure of the other angles.
First draw a picture. When labeling the vertices, the letters are listed, in order.
If , then by the Opposite Angles Theorem.
Example B
Find the values of and .
Remember that opposite sides of a parallelogram are congruent. Set up equations and solve.
Example C
Prove the Opposite Sides Theorem.
Given : is a parallelogram with diagonal
Prove :
Statement | Reason |
---|---|
1. is a parallelogram with diagonal | 1. Given |
2. | 2. Definition of a parallelogram |
3. | 3. Alternate Interior Angles Theorem |
4. | 4. Reflexive PoC |
5. | 5. ASA |
6. | 6. CPCTC |
The proof of the Opposite Angles Theorem is almost identical. You will try this proof in the problem set.
Guided Practice
1. Show that the diagonals of bisect each other.
2. is a parallelogram, and . Solve for .
3. Find the measures of and in the parallelogram below:
Answers:
1. 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.
2. Because this is a parallelogram, the diagonals bisect each other and .
3. Consecutive angles are supplementary so which means that . and are alternate interior angles and since the lines are parallel (since its a parallelogram), that means that .
Practice
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 11-18, find the values 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. | 1. Given |
2. | 2. |
3. | 3. Alternate Interior Angles Theorem |
4. | 4. Reflexive PoC |
5. | 5. |
6. | 6. |
- Parallelogram Diagonals Theorem
Given : is a parallelogram with diagonals and
Prove :
Statement | Reason |
---|---|
1. | 1. |
2. | 2. Definition of a parallelogram |
3. | 3. Alternate Interior Angles Theorem |
4. | 4. |
5. | 5. |
6. | 6. |
- Find and . (The two quadrilaterals with the same side are parallelograms.)