Parallelograms

1. SECTION 6-2
Parallelograms

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?

3. Vocabulary
1. Parallelogram:

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

5. Properties and Theorems
6.3 - Opposite Sides:
6.4 - Opposite Angles:
6.5 - Consecutive Angles:
6.6 - Right Angles:

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:

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:

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:

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 parallelogram has one right angle, then it has four
right angles

10. Properties and Theorems
6.7 - Bisecting Diagonals:
6.8 - Triangles from Diagonals:

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:

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

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

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

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

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

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°

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

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

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

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

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

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

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

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

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

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

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)?

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
⎛
⎝
⎜
⎞
⎠
⎟

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
⎛
⎝⎜
⎞
⎠⎟

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
⎛
⎝⎜
⎞
⎠⎟

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( )

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
⎛
⎝⎜
⎞
⎠⎟

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
⎛
⎝⎜
⎞
⎠⎟

35. 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( )

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.

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

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

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. AB is parallel to CD

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 parallelogram2. AB is parallel to CD

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 ≅ ∠DCA, ∠BDC ≅ ∠DBA
2. AB is parallel to CD

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 ≅ ∠DCA, ∠BDC ≅ ∠DBA
2. AB is parallel to CD
3.Alt. int. ∠’s of parallel lines ≅

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 ≅ ∠DCA, ∠BDC ≅ ∠DBA
2. AB is parallel to CD
3.Alt. int. ∠’s of parallel lines ≅
4. AB ≅ CD

44. 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 ≅ ∠DCA, ∠BDC ≅ ∠DBA
2. AB is parallel to CD
3.Alt. int. ∠’s of parallel lines ≅
4. AB ≅ CD 4. Opposite sides of parallelograms ≅

45. Example 4

46. Example 4
5. ∆APB ≅ ∆CPD

47. Example 4
5. ∆APB ≅ ∆CPD 5.ASA

48. Example 4
5. ∆APB ≅ ∆CPD 5.ASA
6. AP ≅ CP, DP ≅ BP

49. Example 4
5. ∆APB ≅ ∆CPD 5.ASA
6. AP ≅ CP, DP ≅ BP 6. CPCTC

50. Example 4
5. ∆APB ≅ ∆CPD 5.ASA
6. AP ≅ CP, DP ≅ BP 6. CPCTC
7. AC and BD bisect each other

51. 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