6.3: Proving Quadrilaterals are Parallelograms
Learning Objectives
 Prove a quadrilateral is a parallelogram.
 Show a quadrilateral is a parallelogram in the plane.
Review Queue
 Plot the points , and .
 Find the slopes of and . Is a parallelogram?
 Find the point of intersection of the diagonals by finding the midpoint of each.
Know What? You are marking out a baseball diamond and standing at home plate. base is 90 feet away, base is 127.3 feet away, and base is also 90 feet away. The angle at home plate is , from to is . Find the length of the other diagonal (using the Pythagorean Theorem) and determine if the baseball diamond is a parallelogram.
Determining if a Quadrilateral is a Parallelogram
The converses of the theorems in the last section will now be used to see if a quadrilateral is a parallelogram.
Opposite Sides Theorem Converse: If the opposite sides of a quadrilateral are congruent, then the figure is a parallelogram.
If then
Opposite Angles Theorem Converse: If the opposite angles of a quadrilateral are congruent, then the figure is a parallelogram.
If then
Parallelogram Diagonals Theorem Converse: If the diagonals of a quadrilateral bisect each other, then the figure is a parallelogram.
If then
Proof of the Opposite Sides Theorem Converse
Given:
Prove: is a parallelogram
Statement  Reason 

1.  Given 
2.  Reflexive PoC 
3.  SSS 
4.  CPCTC 
5.  Alternate Interior Angles Converse 
6. is a parallelogram  Definition of a parallelogram 
Example 1: Write a twocolumn proof.
Given: , and
Prove: is a parallelogram
Solution:
Statement  Reason 

1. , and  Given 
2.  Alternate Interior Angles 
3.  Reflexive PoC 
4.  SAS 
5.  CPCTC 
6. is a parallelogram  Opposite Sides Converse 
Theorem 610: If a quadrilateral has one set of parallel lines that are also congruent, then it is a parallelogram.
If then
Example 2: Is quadrilateral a parallelogram? How do you know?
Solution:
a) By the Opposite Angles Theorem Converse, is a parallelogram.
b) is not a parallelogram because the diagonals do not bisect each other.
Example 3: Algebra Connection What value of would make a parallelogram?
Solution: . By Theorem 610, would be a parallelogram if .
Showing a Quadrilateral is a Parallelogram in the Plane
To show that a quadrilateral is a parallelogram in the plane, you might need:
 The Slope Formula, .
 The Distance Formula, .
 The Midpoint Formula, .
Example 4: Is the quadrilateral a parallelogram?
Solution: Let’s use Theorem 610 to see if is a parallelogram. First, find the length of and .
Find the slopes.
and the slopes are the same, is a parallelogram.
Example 5: Is the quadrilateral a parallelogram?
Solution: 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.
Know What? Revisited Use the Pythagorean Theorem to find the length of the second diagonal.
The diagonals are equal, so the other two sides of the diamond must also be 90 feet. The baseball diamond is a parallelogram, and more specifically, a square.
Review Questions
 Questions 112 are similar to Example 2.
 Questions 1315 are similar to Example 3.
 Questions 1622 are similar to Examples 4 and 5.
 Questions 2325 are similar to Example 1 and the proof of the Opposite Sides Converse.
For questions 112, determine if the quadrilaterals are parallelograms.
Algebra Connection For questions 1318, determine the value of and that would make the quadrilateral a parallelogram.
For questions 1922, 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.  
2.  
3.  Definition of a quadrilateral 
4.  
5.  Combine Like Terms 
6.  Division PoE 
7. and are supplementary and are supplementary 

8.  Consecutive Interior Angles Converse 
9. is a parallelogram 
 Parallelogram Diagonals Theorem Converse
Given:
Prove: is a parallelogram
Statement  Reason 

1.  
2.  Vertical Angles Theorem 
3.  
4.  
5. is a parallelogram 
 Given: Prove: is a parallelogram
Statement  Reason 

1.  
2.  
3. is a parallelogram 
Review Queue Answers
1.
(a)
is a parallelogram because the opposite sides are parallel.
(b)
Yes, the midpoints of the diagonals are the same, so they bisect each other.