9.2 proving quadrilaterals are parallelograms

9.2 Proving Quadrilaterals are ParallelogramsGeometryMr. Calise

Objectives:

Prove that a quadrilateral is a parallelogram.

Use coordinate geometry with parallelograms.

Theorems

Theorem 9-7: 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.

Ex. 1: Proof of Theorem 6.6Statements:1. AB ≅ CD, AD ≅ CB.2. AC ≅ AC3. ∆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

Theorems

Theorem 9-8: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ABCD is a parallelogram.

A

D

B

C

Theorems

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 9-5: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

ABCD is a parallelogram.

A

D

B

C

Theorems

Theorem 9-6: If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.

A

D

B

C

ABCD is a parallelogram.

Ex. 3: Proof of TheoremGiven: BC║DA, BC ≅ DAProve: ABCD is a

Statements:1. BC ║DA2. DAC ≅ BCA3. AC ≅ AC4. BC ≅ DA5. ∆BAC ≅ ∆DCA6. AB ≅ CD7. ABCD is a

Reasons:1. Given

C

D

B

A

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

Statements:1. BC ║DA2. DAC ≅ BCA3. AC ≅ AC4. BC ≅ DA5. ∆BAC ≅ ∆DCA6. AB ≅ CD7. ABCD is a

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

C

D

B

A

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

Statements:1. BC ║DA2. DAC ≅ BCA3. AC ≅ AC4. BC ≅ DA5. ∆BAC ≅ ∆DCA6. AB ≅ CD7. 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 ≅ DAProve: ABCD is a

Statements:1. BC ║DA2. DAC ≅ BCA3. AC ≅ AC4. BC ≅ DA5. ∆BAC ≅ ∆DCA6. AB ≅ CD7. 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 ≅ DAProve: ABCD is a

Statements:1. BC ║DA2. DAC ≅ BCA3. AC ≅ AC4. BC ≅ DA5. ∆BAC ≅ ∆DCA6. AB ≅ CD7. 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 ≅ DAProve: ABCD is a

Statements:1. BC ║DA2. DAC ≅ BCA3. AC ≅ AC4. BC ≅ DA5. ∆BAC ≅ ∆DCA6. AB ≅ CD7. 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 ≅ DAProve: ABCD is a

Statements:1. BC ║DA2. DAC ≅ BCA3. AC ≅ AC4. BC ≅ DA5. ∆BAC ≅ ∆DCA6. AB ≅ CD7. 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

Ex. 2: Proving Quadrilaterals are Parallelograms

How do we know these opposite sides are parallel?

2.75 in. 2.75 in.

2 in.

2 in.

Ex. 2: Proving Quadrilaterals are Parallelograms

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.

2.75 in. 2.75 in.

2 in.

2 in.

Objective 2: Using Coordinate Geometry

When a figure is in the coordinate plane, you can use the Distance Formula to prove that sides are congruent and you can use the slope formula 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)

Homework

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