Theorems

Theorem 6.6: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

A

D

B

C

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.

A

D

B

C

ABCD is a parallelogram.

Theorems

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

A

D

B

C

ABCD is a parallelogram.

x°

(180 – x)° x°

Theorems

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

ABCD is a parallelogram.

A

D

B

C

Ex. 1: Proof of Theorem 6.6Statements:1. AB CD, AD ≅ ≅

CB.2. AC AC≅3. ∆ABC ≅ ∆CDA4. BAC ≅ DCA,

DAC ≅ BCA5. AB║CD, AD ║CB.6. ABCD is a

Reasons:1. Given

C

D

B

A

Ex. 1: Proof of Theorem 6.6Statements:1. AB CD, AD CB.≅ ≅2. AC AC≅

3. ∆ABC ≅ ∆CDA4. BAC ≅ DCA, DAC

≅ BCA5. AB║CD, AD ║CB.6. ABCD is a

Reasons:1. Given2. Reflexive Prop. of Congruence

C

D

B

A

Ex. 1: Proof of Theorem 6.6Statements:1. AB CD, AD CB.≅ ≅2. AC AC≅

3. ∆ABC ≅ ∆CDA4. BAC ≅ DCA, DAC

≅ BCA5. AB║CD, AD ║CB.6. ABCD is a

Reasons:1. Given2. Reflexive Prop. of Congruence

3. SSS Congruence Postulate

C

D

B

A

Ex. 1: Proof of Theorem 6.6Statements:1. AB CD, AD CB.≅ ≅2. AC AC≅

3. ∆ABC ≅ ∆CDA4. BAC ≅ DCA, DAC

≅ BCA5. AB║CD, AD ║CB.6. ABCD is a

Reasons:1. Given2. Reflexive Prop. of Congruence

3. SSS Congruence Postulate4. CPCTC

C

D

B

A

Ex. 1: Proof of Theorem 6.6Statements:1. AB CD, AD CB.≅ ≅2. AC AC≅

3. ∆ABC ≅ ∆CDA4. BAC ≅ DCA, DAC

≅ BCA5. AB║CD, AD ║CB.6. ABCD is a

Reasons:1. Given2. Reflexive Prop. of Congruence

3. SSS Congruence Postulate4. CPCTC

5. Alternate Interior s Converse

C

D

B

A

Ex. 1: Proof of Theorem 6.6Statements:1. AB CD, AD CB.≅ ≅2. AC AC≅

3. ∆ABC ≅ ∆CDA4. BAC ≅ DCA, DAC

≅ BCA5. AB║CD, AD ║CB.6. ABCD is a

Reasons:1. Given2. Reflexive Prop. of Congruence

3. SSS Congruence Postulate4. CPCTC

5. Alternate Interior s Converse6. Def. of a parallelogram.

C

D

B

A

Ex. 2: Proving Quadrilaterals are Parallelograms

As the sewing box below is opened, the trays are always parallel to each other. Why?

2.75 in. 2.75 in.

2 in.

2 in.

Ex. 2: Proving Quadrilaterals are Parallelograms

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.

2.75 in. 2.75 in.

2 in.

2 in.

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.

A

B C

D

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC DA≅Prove: ABCD is a

Statements:1. BC ║DA2. DAC ≅ BCA3. AC AC≅4. BC DA≅5. ∆BAC ≅ ∆DCA6. AB CD≅7. ABCD is a

Reasons:1. Given

C

D

B

A

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC DA≅Prove: ABCD is a

Statements:1. BC ║DA2. DAC ≅ BCA3. AC AC≅4. BC DA≅5. ∆BAC ≅ ∆DCA6. AB CD≅7. ABCD is a

Reasons:1. Given2. Alt. Int. s Thm.

C

D

B

A

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC DA≅Prove: ABCD is a

Statements:1. BC ║DA2. DAC ≅ BCA3. AC AC≅4. BC DA≅5. ∆BAC ≅ ∆DCA6. AB CD≅7. ABCD is a

Reasons:1. Given2. Alt. Int. s Thm.3. Reflexive Property

C

D

B

A

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC DA≅Prove: ABCD is a

Statements:1. BC ║DA2. DAC ≅ BCA3. AC AC≅4. BC DA≅5. ∆BAC ≅ ∆DCA6. AB CD≅7. ABCD is a

Reasons:1. Given2. Alt. Int. s Thm.3. Reflexive Property4. Given

C

D

B

A

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC DA≅Prove: ABCD is a

Statements:1. BC ║DA2. DAC ≅ BCA3. AC AC≅4. BC DA≅5. ∆BAC ≅ ∆DCA6. AB CD≅7. ABCD is a

Reasons:1. Given2. Alt. Int. s Thm.3. Reflexive Property4. Given5. SAS Congruence

Post.

C

D

B

A

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC DA≅Prove: ABCD is a

Statements:1. BC ║DA2. DAC ≅ BCA3. AC AC≅4. BC DA≅5. ∆BAC ≅ ∆DCA6. AB CD≅7. ABCD is a

Reasons:1. Given2. Alt. Int. s Thm.3. Reflexive Property4. Given5. SAS Congruence

Post.6. CPCTC

C

D

B

A

Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC DA≅Prove: ABCD is a

Statements:1. BC ║DA2. DAC ≅ BCA3. AC AC≅4. BC DA≅5. ∆BAC ≅ ∆DCA6. AB CD≅7. ABCD is a

Reasons:1. Given2. Alt. Int. s Thm.3. Reflexive Property4. Given5. SAS Congruence

Post.6. CPCTC7. If opp. sides of a

quad. are , then it ≅is a .

C

D

B

A

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 parallelograms Show that A(2, -1),

B(1, 3), C(6, 5) and D(7,1) are the vertices of a parallelogram.

6

4

2

-2

-4

5 10 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

Ex. 4: Using properties of parallelograms 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.

6

4

2

-2

-4

5 10 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

Because opposite sides are parallel, ABCD is a parallelogram.

Ex. 4: Using properties of parallelograms 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.

6

4

2

-2

-4

5 10 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

Ex. 4: Using properties of parallelograms 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.

6

4

2

-2

-4

5 10 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)