1. Splash Screen

2. Five-Minute Check (over Lesson 6–1) CCSS Then/Now New Vocabulary
Theorems: Properties of Parallelograms
Proof: Theorem 6.4
Example 1: Real-World Example: Use Properties of Parallelograms
Theorems: Diagonals of Parallelograms
Example 2: Use Properties of Parallelograms and Algebra
Example 3: Parallelograms and Coordinate Geometry
Example 4: Proofs Using the Properties of Parallelograms
Lesson Menu

3. Find the measure of an interior angle of a regular polygon that has 10 sides.
B. 162
C. 144
D. 126
5-Minute Check 1

4. Find the measure of an interior angle of a regular polygon that has 10 sides.
B. 162
C. 144
D. 126
5-Minute Check 1

5. Find the measure of an interior angle of a regular polygon that has 12 sides.
B. 150
C. 165
D. 180
5-Minute Check 2

6. Find the measure of an interior angle of a regular polygon that has 12 sides.
B. 150
C. 165
D. 180
5-Minute Check 2

7. What is the sum of the measures of the interior angles of a 20-gon?
B. 3420
C. 3240
D. 3060
5-Minute Check 3

8. What is the sum of the measures of the interior angles of a 20-gon?
B. 3420
C. 3240
D. 3060
5-Minute Check 3

9. What is the sum of the measures of the interior angles of a 16-gon?
B. 2880
C. 2700
D. 2520
5-Minute Check 4

10. What is the sum of the measures of the interior angles of a 16-gon?
B. 2880
C. 2700
D. 2520
5-Minute Check 4

11. Find x if QRSTU is a regular pentagon.B
C. 12
D. 10
5-Minute Check 5

12. Find x if QRSTU is a regular pentagon.B
C. 12
D. 10
5-Minute Check 5

13. What type of regular polygon has interior angles with a measure of 135°?
A. pentagon
B. hexagon
C. octagon
D. decagon
5-Minute Check 6

14. What type of regular polygon has interior angles with a measure of 135°?
A. pentagon
B. hexagon
C. octagon
D. decagon
5-Minute Check 6

15. G.CO.11 Prove theorems about parallelograms.
Content Standards
G.CO.11 Prove theorems about parallelograms.
G.GPE.4 Use coordinates to prove simple geometric theorems algebraically.
Mathematical Practices
4 Model with mathematics.
3 Construct viable arguments and critique the reasoning of others.
CCSS

16. You classified polygons with four sides as quadrilaterals.
Recognize and apply properties of the sides and angles of parallelograms.
Recognize and apply properties of the diagonals of parallelograms.
Then/Now

17. parallelogram
Vocabulary

18. Concept 1

19. Concept 2

20. Use Properties of Parallelograms
A. CONSTRUCTION In suppose mB = 32, CD = 80 inches, BC = 15 inches. Find AD.
Example 1A

21. AD = BC Opposite sides of a are .
Use Properties of Parallelograms
AD = BC Opposite sides of a are .
= 15 Substitution
Answer:
Example 1

22. AD = BC Opposite sides of a are .
Use Properties of Parallelograms
AD = BC Opposite sides of a are .
= 15 Substitution
Answer: AD = 15 inches
Example 1

23. Use Properties of Parallelograms
B. CONSTRUCTION In suppose mB = 32, CD = 80 inches, BC = 15 inches. Find mC.
Example 1B

24. mC + mB = 180 Cons. s in a are supplementary.
Use Properties of Parallelograms
mC + mB = Cons. s in a are supplementary.
mC + 32 = Substitution
mC = Subtract 32 from each side.
Answer:
Example 1

25. mC + mB = 180 Cons. s in a are supplementary.
Use Properties of Parallelograms
mC + mB = Cons. s in a are supplementary.
mC + 32 = Substitution
mC = Subtract 32 from each side.
Answer: mC = 148
Example 1

26. Use Properties of Parallelograms
C. CONSTRUCTION In suppose mB = 32, CD = 80 inches, BC = 15 inches. Find mD.
Example 1C

27. mD = mB Opp. s of a are . = 32 Substitution Answer:
Use Properties of Parallelograms
mD = mB Opp. s of a are .
= Substitution
Answer:
Example 1

28. mD = mB Opp. s of a are . = 32 Substitution Answer: mD = 32
Use Properties of Parallelograms
mD = mB Opp. s of a are .
= Substitution
Answer: mD = 32
Example 1

29. A. ABCD is a parallelogram. Find AB.
Example 1A

30. A. ABCD is a parallelogram. Find AB.
Example 1A

31. B. ABCD is a parallelogram. Find mC.
Example 1B

32. B. ABCD is a parallelogram. Find mC.
Example 1B

33. C. ABCD is a parallelogram. Find mD.
Example 1C

34. C. ABCD is a parallelogram. Find mD.
Example 1C

35. Concept 3

36. A. If WXYZ is a parallelogram, find the value of r.
Use Properties of Parallelograms and Algebra
A. If WXYZ is a parallelogram, find the value of r.
Opposite sides of a parallelogram are .
Definition of congruence
Substitution
Divide each side by 4.
Answer:
Example 2A

37. A. If WXYZ is a parallelogram, find the value of r.
Use Properties of Parallelograms and Algebra
A. If WXYZ is a parallelogram, find the value of r.
Opposite sides of a parallelogram are .
Definition of congruence
Substitution
Divide each side by 4.
Answer: r = 4.5
Example 2A

38. B. If WXYZ is a parallelogram, find the value of s.
Use Properties of Parallelograms and Algebra
B. If WXYZ is a parallelogram, find the value of s.
8s = 7s + 3 Diagonals of a bisect each other.
s = 3 Subtract 7s from each side.
Answer:
Example 2B

39. B. If WXYZ is a parallelogram, find the value of s.
Use Properties of Parallelograms and Algebra
B. If WXYZ is a parallelogram, find the value of s.
8s = 7s + 3 Diagonals of a bisect each other.
s = 3 Subtract 7s from each side.
Answer: s = 3
Example 2B

40. C. If WXYZ is a parallelogram, find the value of t.
Use Properties of Parallelograms and Algebra
C. If WXYZ is a parallelogram, find the value of t.
ΔWXY  ΔYZW Diagonal separates a parallelogram into  triangles.
YWX  WYZ CPCTC
mYWX = mWYZ Definition of congruence
Example 2C

41. 2t = 18 Substitution t = 9 Divide each side by 2. Answer:
Use Properties of Parallelograms and Algebra
2t = 18 Substitution
t = 9 Divide each side by 2.
Answer:
Example 2C

42. 2t = 18 Substitution t = 9 Divide each side by 2. Answer: t = 9
Use Properties of Parallelograms and Algebra
2t = 18 Substitution
t = 9 Divide each side by 2.
Answer: t = 9
Example 2C

43. A. If ABCD is a parallelogram, find the value of x.
Example 2A

44. A. If ABCD is a parallelogram, find the value of x.
Example 2A

45. B. If ABCD is a parallelogram, find the value of p.
Example 2B

46. B. If ABCD is a parallelogram, find the value of p.
Example 2B

47. C. If ABCD is a parallelogram, find the value of k.
Example 2C

48. C. If ABCD is a parallelogram, find the value of k.
Example 2C

49. Parallelograms and Coordinate Geometry
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)?
Since the diagonals of a parallelogram bisect each other, the intersection point is the midpoint of
Find the midpoint of
Midpoint Formula
Example 3

50. Parallelograms and Coordinate Geometry
Answer:
Example 3

51. Parallelograms and Coordinate Geometry
Answer: The coordinates of the intersection of the diagonals of parallelogram MNPR are (1, 2).
Example 3

52. What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with vertices L(0, –3), M(–2, 1), N(1, 5), O(3, 1)? A. B. C. D.
Example 3

53. What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with vertices L(0, –3), M(–2, 1), N(1, 5), O(3, 1)? A. B. C. D.
Example 3

54. Write a paragraph proof.
Proofs Using the Properties of Parallelograms
Write a paragraph proof.
Given: are diagonals, and point P is the intersection of
Prove: AC and BD bisect each other.
Proof: ABCD is a parallelogram and AC and BD are diagonals; therefore, AB║DC and AC is a transversal. BAC  DCA and ABD  CDB by Theorem 3.2. ΔAPB  ΔCPD by ASA. So, by the properties of congruent triangles BP  DP and AP  CP. Therefore, AC and BD bisect each other.
Example 4

55. To complete the proof below, which of the following is relevant information?
Given: LMNO, LN and MO are diagonals and point Q is the intersection of LN and MO.
Prove: LNO  NLM
A. LO  MN
B. LM║NO
C. OQ  QM
D. Q is the midpoint of LN.
Example 4

56. To complete the proof below, which of the following is relevant information?
Given: LMNO, LN and MO are diagonals and point Q is the intersection of LN and MO.
Prove: LNO  NLM
A. LO  MN
B. LM║NO
C. OQ  QM
D. Q is the midpoint of LN.
Example 4

57. End of the Lesson