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?


Vocabulary

1. Parallelogram:


Vocabulary

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


Properties and Theorems6.3 - Opposite Sides:

6.4 - Opposite Angles:

6.5 - Consecutive Angles:

6.6 - Right Angles:


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

opposite sides are congruent

6.4 - Opposite Angles:


6.6 - Right Angles:




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


6.6 - Right Angles:





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

6.6 - Right Angles:





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


Properties and Theorems6.7 - Bisecting Diagonals:

6.8 - Triangles from Diagonals:


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

diagonals bisect each other

6.8 - Triangles from Diagonals:


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


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



Find each measure.

a. AD 15 inches

b. m∠C

c. m∠D



Find each measure.

a. AD 15 inches

b. m∠C = 180 − 32

c. m∠D



Find each measure.

a. AD 15 inches

b. m∠C = 180 − 32 = 148°

c. m∠D



Find each measure.

a. AD 15 inches

b. m∠C = 180 − 32 = 148°

c. m∠D = 32°


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



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



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



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



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



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



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 = (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



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



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


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 =

x1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜⎞

⎠⎟

M(MP ) =

−3+ 52

,0 + 4

2

⎛⎝⎜

⎞⎠⎟




M =

x1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜⎞

⎠⎟

M(MP ) =

−3+ 52

,0 + 4

2

⎛⎝⎜

⎞⎠⎟ =

22

,42

⎛⎝⎜

⎞⎠⎟




M =

x1+ x

2

2,y

1+ y

2

2

⎛

⎝⎜⎞

⎠⎟

M(MP ) =

−3+ 52

,0 + 4

2

⎛⎝⎜

⎞⎠⎟ =

22

,42

⎛⎝⎜

⎞⎠⎟ = 1, 2( )




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

⎛⎝⎜

⎞⎠⎟




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

⎛⎝⎜

⎞⎠⎟




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( )


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





P is the intersection of AC and BD1. Given






2. AB is parallel to CD






2. Definition of parallelogram2. AB is parallel to CD






2. Definition of parallelogram

3. ∠BAC ≅ ∠ACD, ∠BDC ≅ ∠DBA










3. Alt. int. ∠’s of parallel lines ≅










4. AB ≅ DC










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


Example 4


Example 4

5. ∆APB ≅ ∆CPD


Example 4

5. ∆APB ≅ ∆CPD 5. ASA


Example 4


6. AP ≅ CP, DP ≅ BP


Example 4


6. AP ≅ CP, DP ≅ BP 6. CPCTC


Example 4



7. AC and BD bisect each other


Example 4



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


Check Your Understanding

Review #1-8 on p. 403


Problem Set


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


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Geometry Section 6-2 1112

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