geometry section 6-2 1112
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ParallelogramsTRANSCRIPT
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