2. 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
5. Properties and Theorems
6.3 - Opposite Sides:
6.4 - Opposite Angles:
6.5 - Consecutive Angles:
6.6 - Right Angles:
Tuesday, April 10, 2012
6. Properties and Theorems
6.3 - 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
7. Properties and Theorems
6.3 - 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
8. Properties and Theorems
6.3 - 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
9. Properties and Theorems
6.3 - 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
11. Properties and Theorems
6.7 - Bisecting Diagonals: If a quadrilateral is a parallelogram, then its
diagonals bisect each other
6.8 - Triangles from Diagonals:
Tuesday, April 10, 2012
12. Properties and Theorems
6.7 - 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
13. Example 1
In 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
14. Example 1
In 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
15. Example 1
In 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
16. Example 1
In 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
17. Example 1
In 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
18. Example 2
If 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
19. Example 2
If 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
20. Example 2
If 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
21. Example 2
If 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
22. Example 2
If 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
23. Example 2
If 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
24. Example 2
If 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
25. Example 2
If 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
26. Example 2
If 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
27. Example 2
If 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
28. Example 3
What 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
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟
Tuesday, April 10, 2012
29. Example 3
What 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
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟
M(MP) =
−3+ 5
2
,
0 + 4
2
⎛
⎝⎜
⎞
⎠⎟
Tuesday, April 10, 2012
30. Example 3
What 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
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟
M(MP) =
−3+ 5
2
,
0 + 4
2
⎛
⎝⎜
⎞
⎠⎟ =
2
2
,
4
2
⎛
⎝⎜
⎞
⎠⎟
Tuesday, April 10, 2012
31. Example 3
What 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
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟
M(MP) =
−3+ 5
2
,
0 + 4
2
⎛
⎝⎜
⎞
⎠⎟ =
2
2
,
4
2
⎛
⎝⎜
⎞
⎠⎟ = 1,2( )
Tuesday, April 10, 2012
32. Example 3
What 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
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟
M(MP) =
−3+ 5
2
,
0 + 4
2
⎛
⎝⎜
⎞
⎠⎟ =
2
2
,
4
2
⎛
⎝⎜
⎞
⎠⎟ = 1,2( )
M(NR) =
−1+ 3
2
,
3+1
2
⎛
⎝⎜
⎞
⎠⎟
Tuesday, April 10, 2012
33. Example 3
What 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
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟
M(MP) =
−3+ 5
2
,
0 + 4
2
⎛
⎝⎜
⎞
⎠⎟ =
2
2
,
4
2
⎛
⎝⎜
⎞
⎠⎟ = 1,2( )
M(NR) =
−1+ 3
2
,
3+1
2
⎛
⎝⎜
⎞
⎠⎟ =
2
2
,
4
2
⎛
⎝⎜
⎞
⎠⎟
Tuesday, April 10, 2012
34. Example 3
What 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
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟
M(MP) =
−3+ 5
2
,
0 + 4
2
⎛
⎝⎜
⎞
⎠⎟ =
2
2
,
4
2
⎛
⎝⎜
⎞
⎠⎟ = 1,2( )
M(NR) =
−1+ 3
2
,
3+1
2
⎛
⎝⎜
⎞
⎠⎟ =
2
2
,
4
2
⎛
⎝⎜
⎞
⎠⎟ = 1,2( )
Tuesday, April 10, 2012
35. Example 4
Given 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
36. Example 4
Given 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
37. Example 4
Given 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
1. Given
Tuesday, April 10, 2012
38. Example 4
Given 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
1. Given
2. AB is parallel to CD
Tuesday, April 10, 2012
39. Example 4
Given 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
1. Given
2. Definition of parallelogram2. AB is parallel to CD
Tuesday, April 10, 2012
40. Example 4
Given 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
1. Given
2. Definition of parallelogram
3. ∠BAC ≅ ∠ACD, ∠BDC ≅ ∠DBA
2. AB is parallel to CD
Tuesday, April 10, 2012
41. Example 4
Given 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
1. 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
42. Example 4
Given 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
1. 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
43. Example 4
Given 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
1. 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
