Geometry Section 6-2 1112

SECTION 6-2Parallelograms

Tuesday, April 10, 2012

Essential Questions

How do you recognize and apply properties of the sides and angles of parallelograms?

How do you recognize and apply properties of the diagonals of parallelograms?

Tuesday, April 10, 2012

Vocabulary

1. Parallelogram:

Tuesday, April 10, 2012

Vocabulary

1. Parallelogram: A quadrilateral that has two pairs of parallel sides

Tuesday, April 10, 2012

Properties and Theorems6.3 - Opposite Sides:

6.4 - Opposite Angles:

6.5 - Consecutive Angles:

6.6 - Right Angles:

Tuesday, April 10, 2012

Properties and Theorems6.3 - Opposite Sides: If a quadrilateral is a parallelogram, then its

opposite sides are congruent

6.4 - Opposite Angles:

6.5 - Consecutive Angles:

6.6 - Right Angles:

Tuesday, April 10, 2012

Properties and Theorems6.3 - Opposite Sides: If a quadrilateral is a parallelogram, then its

opposite sides are congruent

6.4 - Opposite Angles: If a quadrilateral is a parallelogram, then its opposite angles are congruent

6.5 - Consecutive Angles:

6.6 - Right Angles:

Tuesday, April 10, 2012

Properties and Theorems6.3 - Opposite Sides: If a quadrilateral is a parallelogram, then its

opposite sides are congruent

6.4 - Opposite Angles: If a quadrilateral is a parallelogram, then its opposite angles are congruent

6.5 - Consecutive Angles: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary

6.6 - Right Angles:

Tuesday, April 10, 2012

Properties and Theorems6.3 - Opposite Sides: If a quadrilateral is a parallelogram, then its

opposite sides are congruent

6.4 - Opposite Angles: If a quadrilateral is a parallelogram, then its opposite angles are congruent

6.5 - Consecutive Angles: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary

6.6 - Right Angles: If a quadrilateral has one right angle, then it has four right angles

Tuesday, April 10, 2012

Properties and Theorems6.7 - Bisecting Diagonals:

6.8 - Triangles from Diagonals:

Tuesday, April 10, 2012

Properties and Theorems6.7 - Bisecting Diagonals: If a quadrilateral is a parallelogram, then its

diagonals bisect each other

6.8 - Triangles from Diagonals:

Tuesday, April 10, 2012

Properties and Theorems6.7 - Bisecting Diagonals: If a quadrilateral is a parallelogram, then its

diagonals bisect each other

6.8 - Triangles from Diagonals: If a quadrilateral is a parallelogram, then each diagonal separates the parallelogram into two congruent triangles

Tuesday, April 10, 2012

Example 1In ▱ABCD, suppose m∠B = 32°, CD = 80 inches, and BC = 15 inches.

Find each measure.

a. AD

b. m∠C

c. m∠D

Tuesday, April 10, 2012

Example 1In ▱ABCD, suppose m∠B = 32°, CD = 80 inches, and BC = 15 inches.

Find each measure.

a. AD 15 inches

b. m∠C

c. m∠D

Tuesday, April 10, 2012

Example 1In ▱ABCD, suppose m∠B = 32°, CD = 80 inches, and BC = 15 inches.

Find each measure.

a. AD 15 inches

b. m∠C = 180 − 32

c. m∠D

Tuesday, April 10, 2012

Example 1In ▱ABCD, suppose m∠B = 32°, CD = 80 inches, and BC = 15 inches.

Find each measure.

a. AD 15 inches

b. m∠C = 180 − 32 = 148°

c. m∠D

Tuesday, April 10, 2012

Example 1In ▱ABCD, suppose m∠B = 32°, CD = 80 inches, and BC = 15 inches.

Find each measure.

a. AD 15 inches

b. m∠C = 180 − 32 = 148°

c. m∠D = 32°

Tuesday, April 10, 2012

Example 2If WXYZ is a parallelogram, find the value of the indicated variable.

m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°,

WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18

a. r b. s

c. t

Tuesday, April 10, 2012

Example 2If WXYZ is a parallelogram, find the value of the indicated variable.

m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°,

WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18

a. r

WX = ZY

b. s

c. t

Tuesday, April 10, 2012

Example 2If WXYZ is a parallelogram, find the value of the indicated variable.

m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°,

WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18

a. r

WX = ZY

4r = 18

b. s

c. t

Tuesday, April 10, 2012

Example 2If WXYZ is a parallelogram, find the value of the indicated variable.

m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°,

WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18

a. r

WX = ZY

4r = 18

r = 4.5

b. s

c. t

Tuesday, April 10, 2012

Example 2If WXYZ is a parallelogram, find the value of the indicated variable.

m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°,

WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18

a. r

WX = ZY

4r = 18

r = 4.5

b. s

ZV = VX c. t

Tuesday, April 10, 2012

Example 2If WXYZ is a parallelogram, find the value of the indicated variable.

m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°,

WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18

a. r

WX = ZY

4r = 18

r = 4.5

b. s

ZV = VX

8s = 7s + 3

c. t

Tuesday, April 10, 2012

Example 2If WXYZ is a parallelogram, find the value of the indicated variable.

m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°,

WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18

a. r

WX = ZY

4r = 18

r = 4.5

b. s

ZV = VX

8s = 7s + 3

s = 3

c. t

Tuesday, April 10, 2012

Example 2If WXYZ is a parallelogram, find the value of the indicated variable.

m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°,

WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18

a. r

WX = ZY

4r = 18

r = 4.5

b. s

ZV = VX

8s = 7s + 3

s = 3

c. t

m∠VWX = m∠VYZ

Tuesday, April 10, 2012

Example 2If WXYZ is a parallelogram, find the value of the indicated variable.

m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°,

WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18

a. r

WX = ZY

4r = 18

r = 4.5

b. s

ZV = VX

8s = 7s + 3

s = 3

c. t

m∠VWX = m∠VYZ

2t = 18

Tuesday, April 10, 2012

Example 2If WXYZ is a parallelogram, find the value of the indicated variable.

m∠VWX = (2t)°, m∠VYX = 40°, m∠VYZ =18°,

WX = 4r, ZV = 8s, VX = 7s + 3, ZY =18

a. r

WX = ZY

4r = 18

r = 4.5

b. s

ZV = VX

8s = 7s + 3

s = 3

c. t

m∠VWX = m∠VYZ

2t = 18 t = 9

Tuesday, April 10, 2012

Example 3What are the coordinates of the intersection of the diagonals of

parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)?

M =

x1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜⎞

⎠⎟

Tuesday, April 10, 2012

Example 3What are the coordinates of the intersection of the diagonals of

parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)?

M =

x1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜⎞

⎠⎟

M(MP ) =

−3+ 52

,0 + 4

2

⎛⎝⎜

⎞⎠⎟

Tuesday, April 10, 2012

Example 3What are the coordinates of the intersection of the diagonals of

parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)?

M =

x1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜⎞

⎠⎟

M(MP ) =

−3+ 52

,0 + 4

2

⎛⎝⎜

⎞⎠⎟ =

22

,42

⎛⎝⎜

⎞⎠⎟

Tuesday, April 10, 2012

Example 3What are the coordinates of the intersection of the diagonals of

parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)?

M =

x1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜⎞

⎠⎟

M(MP ) =

−3+ 52

,0 + 4

2

⎛⎝⎜

⎞⎠⎟ =

22

,42

⎛⎝⎜

⎞⎠⎟ = 1, 2( )

Tuesday, April 10, 2012

Example 3What are the coordinates of the intersection of the diagonals of

parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)?

M =

x1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜⎞

⎠⎟

M(MP ) =

−3+ 52

,0 + 4

2

⎛⎝⎜

⎞⎠⎟ =

22

,42

⎛⎝⎜

⎞⎠⎟ = 1, 2( )

M(NR) =

−1+ 32

,3+1

2

⎛⎝⎜

⎞⎠⎟

Tuesday, April 10, 2012

Example 3What are the coordinates of the intersection of the diagonals of

parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)?

M =

x1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜⎞

⎠⎟

M(MP ) =

−3+ 52

,0 + 4

2

⎛⎝⎜

⎞⎠⎟ =

22

,42

⎛⎝⎜

⎞⎠⎟ = 1, 2( )

M(NR) =

−1+ 32

,3+1

2

⎛⎝⎜

⎞⎠⎟

=22

,42

⎛⎝⎜

⎞⎠⎟

Tuesday, April 10, 2012

Example 3What are the coordinates of the intersection of the diagonals of

parallelogram MNPR with vertices M(−3, 0), N(−1, 3), P(5, 4) , and R(3, 1)?

M =

x1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜⎞

⎠⎟

M(MP ) =

−3+ 52

,0 + 4

2

⎛⎝⎜

⎞⎠⎟ =

22

,42

⎛⎝⎜

⎞⎠⎟ = 1, 2( )

M(NR) =

−1+ 32

,3+1

2

⎛⎝⎜

⎞⎠⎟

=22

,42

⎛⎝⎜

⎞⎠⎟ = 1, 2( )

Tuesday, April 10, 2012

Example 4Given ▱ABCD, AC and BD are diagonals, P is the intersection of AC

and BD, prove that AC and BD bisect each other.

Tuesday, April 10, 2012

Example 4Given ▱ABCD, AC and BD are diagonals, P is the intersection of AC

and BD, prove that AC and BD bisect each other.

1. ABCD is a parallelogram, AC and BD are diagonals,

P is the intersection of AC and BD

Tuesday, April 10, 2012

Example 4Given ▱ABCD, AC and BD are diagonals, P is the intersection of AC

and BD, prove that AC and BD bisect each other.

1. ABCD is a parallelogram, AC and BD are diagonals,

P is the intersection of AC and BD1. Given

Tuesday, April 10, 2012

Example 4Given ▱ABCD, AC and BD are diagonals, P is the intersection of AC

and BD, prove that AC and BD bisect each other.

1. ABCD is a parallelogram, AC and BD are diagonals,

P is the intersection of AC and BD1. Given

2. AB is parallel to CD

Tuesday, April 10, 2012

Example 4Given ▱ABCD, AC and BD are diagonals, P is the intersection of AC

and BD, prove that AC and BD bisect each other.

1. ABCD is a parallelogram, AC and BD are diagonals,

P is the intersection of AC and BD1. Given

2. Definition of parallelogram2. AB is parallel to CD

Tuesday, April 10, 2012

Example 4Given ▱ABCD, AC and BD are diagonals, P is the intersection of AC

and BD, prove that AC and BD bisect each other.

1. ABCD is a parallelogram, AC and BD are diagonals,

P is the intersection of AC and BD1. Given

2. Definition of parallelogram

3. ∠BAC ≅ ∠ACD, ∠BDC ≅ ∠DBA

2. AB is parallel to CD

Tuesday, April 10, 2012

Example 4Given ▱ABCD, AC and BD are diagonals, P is the intersection of AC

and BD, prove that AC and BD bisect each other.

1. ABCD is a parallelogram, AC and BD are diagonals,

P is the intersection of AC and BD1. Given

2. Definition of parallelogram

3. ∠BAC ≅ ∠ACD, ∠BDC ≅ ∠DBA

2. AB is parallel to CD

3. Alt. int. ∠’s of parallel lines ≅

Tuesday, April 10, 2012

Example 4Given ▱ABCD, AC and BD are diagonals, P is the intersection of AC

and BD, prove that AC and BD bisect each other.

1. ABCD is a parallelogram, AC and BD are diagonals,

P is the intersection of AC and BD1. Given

2. Definition of parallelogram

3. ∠BAC ≅ ∠ACD, ∠BDC ≅ ∠DBA

2. AB is parallel to CD

3. Alt. int. ∠’s of parallel lines ≅

4. AB ≅ DC

Tuesday, April 10, 2012

Example 4Given ▱ABCD, AC and BD are diagonals, P is the intersection of AC

and BD, prove that AC and BD bisect each other.

1. ABCD is a parallelogram, AC and BD are diagonals,

P is the intersection of AC and BD1. Given

2. Definition of parallelogram

3. ∠BAC ≅ ∠ACD, ∠BDC ≅ ∠DBA

2. AB is parallel to CD

3. Alt. int. ∠’s of parallel lines ≅

4. AB ≅ DC 4. Opposite sides of parallelograms ≅

Tuesday, April 10, 2012

Example 4

Tuesday, April 10, 2012

Example 4

5. ∆APB ≅ ∆CPD

Tuesday, April 10, 2012

Example 4

5. ∆APB ≅ ∆CPD 5. ASA

Tuesday, April 10, 2012

Example 4

5. ∆APB ≅ ∆CPD 5. ASA

6. AP ≅ CP, DP ≅ BP

Tuesday, April 10, 2012

Example 4

5. ∆APB ≅ ∆CPD 5. ASA

6. AP ≅ CP, DP ≅ BP 6. CPCTC

Tuesday, April 10, 2012

Example 4

5. ∆APB ≅ ∆CPD 5. ASA

6. AP ≅ CP, DP ≅ BP 6. CPCTC

7. AC and BD bisect each other

Tuesday, April 10, 2012

Example 4

5. ∆APB ≅ ∆CPD 5. ASA

6. AP ≅ CP, DP ≅ BP 6. CPCTC

7. AC and BD bisect each other 7. Def. of bisect

Tuesday, April 10, 2012

Check Your Understanding

Review #1-8 on p. 403

Tuesday, April 10, 2012

Problem Set

Tuesday, April 10, 2012

Problem Set

p. 403 #9-23 odd, 27-35 odd, 43, 51, 57

“The best way to predict the future is to invent it.” - Alan Kay

Tuesday, April 10, 2012