Chapter 6
Section 6.3
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Quick Review Some Properties of Parallelograms
Definition : A parallelogram is a quadrilateral with both pairs of opposite sides congruent
If a quadrilateral is a parallelogram, then …
Theorem 6.2
Theorem 6.3
Theorem 6.4
Theorem 6.5
both pairs of opposite sides are congruent both pairs of opposite angles are congruent consecutive angles are supplementary diagonals bisect each other
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Proving a Quad is a Parallelogram
Theorem
Theorem 6.6
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram
Quadrilateral PQRS with both pairs of opposite sides congruent
PQRS is a parallelogram
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Proving a Quad is a Parallelogram
Theorem
Theorem 6.7
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram
Quadrilateral PQRS with both pairs of opposite angles congruent
PQRS is a parallelogram
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Proving a Quad is a Parallelogram
Theorem
Theorem 6.8
If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram m

S + m

P = 180 and m

S + m

R = 180
PQRS is a parallelogram
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Proving a Quad is a Parallelogram
Theorem
Theorem 6.9
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
PR bi sec ts QS and QS bi sec ts PR
PQRS is a parallelogram
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Proving a Quad is a Parallelogram
Theorem
Theorem 6.10
If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram
PQ // SR and PQ

SR
PQRS is a parallelogram
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Proving a Quad is a Parallelogram
Yes, one pair of opposite sides congruent and parallel
Yes, diagonals bisect each other
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Proving a Quad is a Parallelogram
Yes, both pairs of opposite sides congruent
No, same pair of opposite sides must be parallel and congruent
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Proving a Quad is a Parallelogram
Yes, could show both pairs of opposite sides are parallel
Yes, Consecutive angles are supplementary
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Proving a Quad is a Parallelogram
Theorem 6.9: AE

EC
Theorem 6.8: m

CDA + m

DCB = 180
OR m

DAB + m

ABC = 180
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Proving a Quad is a Parallelogram
For a Quadrilateral to be a parallelogram opposite sides must be congruent
3x = 6 x = 2 x + 2 = y - 1
2 + 2 = y – 1
4 = y – 1
5 = y
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Proving a Quad is a Parallelogram
For a Quadrilateral to be a parallelogram opposite angles must be congruent
2x = 70 x = 35
3x + 5 = x + 3y
3(35) + 5 = 35 + 3y
110 = 35 + 3y
75 = 3y
25 = y
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Proving a Quad is a Parallelogram
For a Quadrilateral to be a parallelogram diagonals must bisect each other
3x = 12 x = 4 x + y = 5y
4 + y = 5y
4 = 4y
1 = y
Proving Quadrilaterals are Parallelograms
Lesson 6.3

Use both slope and distance formula to show one pair of opposite side is congruent and parallel
• Show that the quadrilateral with vertices A(-1, -2),
B(5,3), C(-6,2), and D(0,7) is a parallelogram
( 5
 
1 ) 2
AB = CD

( 3
 
2 ) 2 
(

6

0 ) 2 
( 2

7 ) 2
( 6 )
2 
( 5 )
2 
(

6 )
2 
(

5 )
2
36

25

36

25
61

61

3
AB // CD
2
5
 
1
3

2
5

1
//
//
7

0
 
6
7

2
0

6
2
5
6
//
5
6

Proving Quadrilaterals are Parallelograms

Lesson 6.3

HW
Pg 342 #’s 9, 10, 12, 17-19 all, & 25-26
Quiz on Thursday!!

Proving Quadrilaterals are Parallelograms
Lesson 6.3