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6.3 Proving Quadrilaterals are Parallelograms. Objectives/Assignment. Prove that a quadrilateral is a parallelogram. Use coordinate geometry with parallelograms. Assignment: 1-26 all, 45-47 all, quiz page 346.

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objectives assignment
Objectives/Assignment
  • Prove that a quadrilateral is a parallelogram.
  • Use coordinate geometry with parallelograms.
  • Assignment: 1-26 all, 45-47 all, quiz page 346
four more theorems converses
Theorem 6.6: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.FOUR More Theorems “Converses”

ABCD is a parallelogram.

theorems
Theorem 6.7: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.Theorems

ABCD is a parallelogram.

theorems1
Theorem 6.8: If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.Theorems

(180 – x)°

ABCD is a parallelogram.

theorems2
Theorem 6.9: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.Theorems

ABCD is a parallelogram.

ex 1 proof of theorem 6 6
Statements:

AB ≅ CD, AD ≅ CB.

AC ≅ AC

∆ABC ≅ ∆CDA

BAC ≅ DCA, DAC ≅ BCA

AB║CD, AD ║CB.

ABCD is a 

Reasons:

Given

Ex. 1: Proof of Theorem 6.6
ex 1 proof of theorem 6 61
Statements:

AB ≅ CD, AD ≅ CB.

AC ≅ AC

∆ABC ≅ ∆CDA

BAC ≅ DCA, DAC ≅ BCA

AB║CD, AD ║CB.

ABCD is a 

Reasons:

Given

Reflexive Prop. of Congruence

Ex. 1: Proof of Theorem 6.6
ex 1 proof of theorem 6 62
Statements:

AB ≅ CD, AD ≅ CB.

AC ≅ AC

∆ABC ≅ ∆CDA

BAC ≅ DCA, DAC ≅ BCA

AB║CD, AD ║CB.

ABCD is a 

Reasons:

Given

Reflexive Prop. of Congruence

SSS Congruence Postulate

Ex. 1: Proof of Theorem 6.6
ex 1 proof of theorem 6 63
Statements:

AB ≅ CD, AD ≅ CB.

AC ≅ AC

∆ABC ≅ ∆CDA

BAC ≅ DCA, DAC ≅ BCA

AB║CD, AD ║CB.

ABCD is a 

Reasons:

Given

Reflexive Prop. of Congruence

SSS Congruence Postulate

CPOCTAC

Ex. 1: Proof of Theorem 6.6
ex 1 proof of theorem 6 64
Statements:

AB ≅ CD, AD ≅ CB.

AC ≅ AC

∆ABC ≅ ∆CDA

BAC ≅ DCA, DAC ≅ BCA

AB║CD, AD ║CB.

ABCD is a 

Reasons:

Given

Reflexive Prop. of Congruence

SSS Congruence Postulate

CPOCTAC

Alternate Interior s Converse

Ex. 1: Proof of Theorem 6.6
ex 1 proof of theorem 6 65
Statements:

AB ≅ CD, AD ≅ CB.

AC ≅ AC

∆ABC ≅ ∆CDA

BAC ≅ DCA, DAC ≅ BCA

AB║CD, AD ║CB.

ABCD is a 

Reasons:

Given

Reflexive Prop. of Congruence

SSS Congruence Postulate

CPOCTAC

Alternate Interior s Converse

Def. of a parallelogram.

Ex. 1: Proof of Theorem 6.6
ex 2 proving quadrilaterals are parallelograms
Ex. 2: Proving Quadrilaterals are Parallelograms
  • As the sewing box below is opened, the trays are always parallel to each other. Why?
ex 2 proving quadrilaterals are parallelograms1
Each pair of hinges are opposite sides of a quadrilateral. The 2.75 inch sides of the quadrilateral are opposite and congruent. The 2 inch sides are also opposite and congruent. Because opposite sides of the quadrilateral are congruent, it is a parallelogram. By the definition of a parallelogram, opposite sides are parallel, so the trays of the sewing box are always parallel.Ex. 2: Proving Quadrilaterals are Parallelograms
another theorem
Another Theorem ~
  • Theorem 6.10—If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.
  • ABCD is a

parallelogram.

B

C

A

D

Box on bottom of page 340

ex 3 proof of theorem 6 10 given bc da bc da prove abcd is a
Statements:

BC ║DA

DAC ≅ BCA

AC ≅ AC

BC ≅ DA

∆BAC ≅ ∆DCA

AB ≅ CD

ABCD is a 

Reasons:

1. Given

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 
ex 3 proof of theorem 6 10 given bc da bc da prove abcd is a1
Statements:

BC ║DA

DAC ≅ BCA

AC ≅ AC

BC ≅ DA

∆BAC ≅ ∆DCA

AB ≅ CD

ABCD is a 

Reasons:

Given

Alt. Int. s Thm.

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 
ex 3 proof of theorem 6 10 given bc da bc da prove abcd is a2
Statements:

BC ║DA

DAC ≅ BCA

AC ≅ AC

BC ≅ DA

∆BAC ≅ ∆DCA

AB ≅ CD

ABCD is a 

Reasons:

Given

Alt. Int. s Thm.

Reflexive Property

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 
ex 3 proof of theorem 6 10 given bc da bc da prove abcd is a3
Statements:

BC ║DA

DAC ≅ BCA

AC ≅ AC

BC ≅ DA

∆BAC ≅ ∆DCA

AB ≅ CD

ABCD is a 

Reasons:

Given

Alt. Int. s Thm.

Reflexive Property

Given

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 
ex 3 proof of theorem 6 10 given bc da bc da prove abcd is a4
Statements:

BC ║DA

DAC ≅ BCA

AC ≅ AC

BC ≅ DA

∆BAC ≅ ∆DCA

AB ≅ CD

ABCD is a 

Reasons:

Given

Alt. Int. s Thm.

Reflexive Property

Given

SAS Congruence Post.

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 
ex 3 proof of theorem 6 10 given bc da bc da prove abcd is a5
Statements:

BC ║DA

DAC ≅ BCA

AC ≅ AC

BC ≅ DA

∆BAC ≅ ∆DCA

AB ≅ CD

ABCD is a 

Reasons:

Given

Alt. Int. s Thm.

Reflexive Property

Given

SAS Congruence Post.

CPOCTAC

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 
ex 3 proof of theorem 6 10 given bc da bc da prove abcd is a6
Statements:

BC ║DA

DAC ≅ BCA

AC ≅ AC

BC ≅ DA

∆BAC ≅ ∆DCA

AB ≅ CD

ABCD is a 

Reasons:

Given

Alt. Int. s Thm.

Reflexive Property

Given

SAS Congruence Post.

CPOCTAC

If opp. sides of a quad. are ≅, then it is a .

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 
objective 2 using coordinate geometry
Objective 2: Using Coordinate Geometry
  • When a figure is in the coordinate plane, you can use the Distance Formula (see—it never goes away) to prove that sides are congruent and you can use the slope formula (see how you use this again?) to prove sides are parallel.
ex 4 using properties of parallelograms1
Method 1—Show that opposite sides have the same slope, so they are parallel.

Slope of AB.

3-(-1) = - 4

1 - 2

Slope of CD.

1 – 5 = - 4

7 – 6

Slope of BC.

5 – 3 = 2

6 - 1 5

Slope of DA.

- 1 – 1 = 2

2 - 7 5

AB and CD have the same slope, so they are parallel. Similarly, BC ║ DA.

Ex. 4: Using properties of parallelograms

Because opposite sides are parallel, ABCD is a parallelogram.

ex 4 using properties of parallelograms2
Method 2—Show that opposite sides have the same length.

AB=√(1 – 2)2 + [3 – (- 1)2] = √17

CD=√(7 – 6)2 + (1 - 5)2 = √17

BC=√(6 – 1)2 + (5 - 3)2 = √29

DA= √(2 – 7)2 + (-1 - 1)2 = √29

AB ≅ CD and BC ≅ DA. Because both pairs of opposites sides are congruent, ABCD is a parallelogram.

Ex. 4: Using properties of parallelograms
ex 4 using properties of parallelograms3
Method 3—Show that one pair of opposite sides is congruent and parallel.

Slope of AB = Slope of CD = -4

AB=CD = √17

AB and CD are congruent and parallel, so ABCD is a parallelogram.

Ex. 4: Using properties of parallelograms