1. Warm – up 11/07 MAKE SURE YOU ARE SITTING IN YOUR NEW SEAT!! Start a new sheet for Ch. 5 warm-ups Answer the following –Come up with your own definition of a quadrilateral (Remember a quad is a polygon) –What is the internal angle sum of a quadrilateral?

2. Chapter 5 Quadrilaterals Apply the definition of a parallelogram Prove that certain quadrilaterals are parallelograms Apply the theorems and definitions about the special quadrilaterals

3. 5-1 Properties of Parallelograms Objectives Apply the definition of a parallelogram List the other properties of a parallelogram through new theorems

4. Quadrilaterals Any 4 sided figure

5. He’s Back…. Parallelograms are special types of quadrilaterals with unique properties If you know you have a parallelogram, then you can prove that these unique properties exist… With each property we learn, say the following to yourself.. –“If a quadrilateral is a parallelogram, then _____________.”

6. Definition of a Parallelogram ( ) If a quadrilateral is a parallelogram, then both pairs of opposite sides are parallel. ABCD AB C D Partners: What do we know about the angles of a parallelogram b/c it has parallel sides? What do you think the definition is based on the diagram?

7. Naming a Parallelogram Use the symbol for parallelogram and name using the 4 vertices in order either clockwise or counter clockwise. ABCD AB C D

8. The fact that we know opposite sides are parallel, we can deduce addition properties through theorems… AB C D

9. Opposite sides of a parallelogram are congruent. Theorem AB C DWhat would be our plan for solving this theorem?

10. Opposite angles of a parallelogram are congruent. Theorem AB C D What did we discuss at the beginning of the lesson about s-s int. angles?

11. The diagonals of a parallelogram bisect each other. Theorem AB C DWhat is another name for AC and BD?

12. Parallelograms: What we now know… If a quad is a parallelogram, then… From the definition.. 1.Both pairs of opposite sides are parallel From theorems… 1.Both pairs of opposite sides are congruent 2.Both pairs of opposite angles are congruent 3.The diagonals of a parallelogram bisect each other

13. Remote Time True or False

14. Every parallelogram is a quadrilateral

15. True or False Every quadrilateral is a parallelogram

16. True or False All angles of a parallelogram are congruent

17. True or False All sides of a parallelogram are congruent

18. True or False In RSTU, RS | |TU.  Hint draw a picture

19. True or False In ABCD, if m  A = 50, then m  C = 130.  Hint draw a picture

20. True or False In XWYZ, XY  WZ  Hint draw a picture

21. True or False In ABCD, AC and BD bisect each other  Hint draw a picture

22. White Board Practice Given ABCD Name all pairs of parallel sides AB || DC BC || AD

23. White Board Practice Given ABCD Name all pairs of congruent angles Draw the parallelogram with the diagonals intersecting at E Given ABCD  BAD   DCB  CBD   ADB  ABC   CDA  ABD   CDB  BEA   DEC  BCA   DAC  BEC   DEA  BAC   DCA

24. White Board Practice Given ABCD Name all pairs of congruent segments Draw the parallelogram with the diagonals intersecting at E

25. White Board Practice Given ABCD AB  CD BC  DA BE  ED AE  EC

26. Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. White Board Groups a R U T S 9 b 6 yºyº 80º xºxº x = 80 y = 100 a = 6 b = 9

27. Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. White Board Groups a R U T S 9 b 12 yºyº 35º xºxº 45º x = 100 y = 45 a = 12 b = 9

28. Given this parallelogram with the diagonals drawn. White Board Groups 18 2x + 8 4y - 2 22 x = 5 y = 6

29. 5-2:Ways to Prove that Quadrilaterals are Parallelograms Objectives Learn about ways to prove a quadrilateral is a parallelogram

30. What we already know… If a quad is a parallelogram, then… –4 properties What we are going to learn.. –What if we don’t know if a quad is a parallelogram, how can we prove that it is one?

31. In the courtroom… “ What we have ladies and gentlemen is a quadrilateral, and I believe this particular quadrilateral happens to be a parallelogram, and I have the evidence to prove it!”

32. Use the Definition of a Parallelogram If both pairs of opposite sides of a quadrilateral are parallel then… the quadrilateral is a parallelogram. AB C D The definition is a biconditional, so it can be used either way.

33. Theorem If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram. Show that both pairs of opposite sides are congruent. A B C D

34. Theorem If one pair of opposite sides of a quadrilateral are both congruent and parallel, then it is a parallelogram. Show that one pair of opposite sides are both congruent and parallel. A B C D

35. Theorem If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram. Show that both pairs of opposite angles are congruent. AB C D

36. Theorem If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. Show that the diagonals bisect each other. AB C D X

37. Five ways to prove a Quadrilateral is a Parallelogram 1.Show that both pairs of opposite sides parallel 2.Show that both pairs of opposite sides congruent 3.Show that one pair of opposite sides are both congruent and parallel 4.Show that both pairs of opposite angles congruent 5.Show that diagonals that bisect each other

38. The diagonals of a quadrilateral _____________ bisect each other A. Sometimes B.Always C.Never D.I don’t know

39. If the measure of two angles of a quadrilateral are equal, then the quadrilateral is ____________ a parallelogram A)Sometimes B)Always C)Never D)I don’t know

40. If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is ___________ a parallelogram A. Sometimes B. Always C. Never D. I don’t know

41. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is __________ a parallelogram A.) Sometimes B.) Always C.) Never D.) I don’t know

42. To prove a quadrilateral is a parallelogram, it is ________ enough to show that one pair of opposite sides is parallel. A.) Sometimes B.) Always C.) Never D.) I don’t know

43. Whiteboards Open book to page 173 –Answer the following… #2 #3 #6 #9

44. 5-3 Theorems Involving Parallel Lines Objectives Apply the theorems about parallel lines and triangles

45. Theorem If two lines are parallel, then all points on one line are equidistant from the other line. m n How do we measure the distance from a point to a line? What does equidistant mean? Demo: 6 volunteers

46. Theorem If three parallel lines cut off congruent segments on one transversal, then they do so on any transversal. A B C D E F Demo: Popsicle Sticks and lined paper

47. Theorem A line that contains the midpoint of one side of a triangle and is parallel to a another side passes through the midpoint of the third side. A BC XY 1 2 3 1 2 3

48. Theorem A segment that joins the midpoints of two sides of a triangle is parallel to the third side and its length is half the length of the third side. A BC XY 1 2 3 12 3 If BC is 12 then XY =?

49. White Board Practice Given: R, S, and T are midpoint of the sides of  ABC C A B T R S ABBCACSTRTRS 121418 152210 97.8

50. White Board Practice Given: R, S, and T are midpoint of the sides of  ABC C A B T R S ABBCACSTRTRS 121418679 201522107.511 1018 15.6 597.8

51. ST is parallel to what side? BC is parallel to what side? A B T R S C

52. White Board Practice Given that AR | | BS | | CT; RS  ST A B C T S R

53. White Board Practice Given that AR | | BS | | CT; RS  ST If RS = 12, then ST = ____ A B C T S R

54. White Board Practice Given that AR | | BS | | CT; RS  ST If RS = 12, then ST = 12 A B C T S R

55. White Board Practice Given that AR | | BS | | CT; RS  ST If AB = 8, then BC = ___ A B C T S R

56. White Board Practice Given that AR | | BS | | CT; RS  ST If AB = 8, then BC = 8 A B C T S R

57. White Board Practice Given that AR | | BS | | CT; RS  ST If AC = 20, then AB = ___ A B C T S R

58. White Board Practice Given that AR | | BS | | CT; RS  ST If AC = 20, then AB = 10 A B C T S R

59. White Board Practice Given that AR | | BS | | CT; RS  ST If AC = 10x, then BC =____ A B C T S R

60. White Board Practice Given that AR | | BS | | CT; RS  ST If AC = 10x, then BC = 5x A B C T S R

61. QUIZ REVIEW 1.Know the properties of a parallelogram 2.Know the 5 ways to prove a quad is a parallelogram 1.“Show that….. 3.Solve problems using the theorems from section 5-3

62. 5.4 Special Parallelograms Objectives Apply the definitions and identify the special properties of a rectangle, rhombus and square.

63. Parallelograms: What we now know… From the definition.. 1.Both pairs of opposite sides are parallel From theorems… 1.Both pairs of opposite sides are congruent 2.Both pairs of opposite angles are congruent 3.The diagonals of a parallelogram bisect each other

64. QUADRILATERALS parallelogram Square Rhombus Rectangle

65. By definition, it is a quadrilateral with four right angles. R S T V

66. Rhombus By definition, it is a quadrilateral with four congruent sides. A B C D

67. Square By definition, it is a quadrilateral with four right angles and four congruent sides. A B C D The square is the most specific type of quadrilateral. What do you notice about the definition compared to the previous two?

68. Theorem The diagonals of a rectangle are congruent. WY  XZ W XY Z P What can we conclude about the smaller segments that make up the diagonals?

69. Finding the special properties of a Rhombus Apply the properties of a parallelogram to find 2 special properties that apply to the Rhombus. Hint: both properties involve angles.

70. Theorem The diagonals of a rhombus are perpendicular. J K L M X What does the definition of perpendicular lines tell us?

71. Theorem Each diagonal of a rhombus bisects the opposite angles. J K L M X

72. Theorem If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle. R S T V WHY?????????

73. Theorem If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. A B C D WHY?

74. X ZY M

75. Theorem The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. X ZY M

76. White Board Practice Quadrilateral ABCD is a rhombus Find the measure of each angle 1.  ACD 2.  DEC 3.  EDC 4.  ABC D A B C E 62º

77. White Board Practice Quadrilateral ABCD is a rhombus Find the measure of each angle 1.  ACD = 62 2.  DEC = 90 3.  EDC = 28 4.  ABC = 56 D A B C E 62º

78. White Board Practice Quadrilateral MNOP is a rectangle Find the measure of each angle 1. m  PON = 2. m  PMO = 3. PL = 4. MO = P MN O L 29º 12

79. White Board Practice Quadrilateral MNOP is a rectangle Find the measure of each angle 1. m  PON = 90 2. m  PMO = 61 3. PL = 12 4. MO = 24 P MN O L 29º 12

80. White Board Practice  ABC is a right  ; M is the midpoint of AB 1. If AM = 7, then MB = ____, AB = ____, and CM = _____. C A B M

81. White Board Practice  ABC is a right  ; M is the midpoint of AB 1. If AM = 7, then MB = 7, AB = 14, and CM = 7. C A B M

82. White Board Practice  ABC is a right  ; M is the midpoint of AB 1. If AB = x, then AM = ____, MB = _____, and MC = _____. C A B M

83. White Board Practice  ABC is a right  ; M is the midpoint of AB 1. If AB = x, then AM = ½ x, MB = ½ x, and MC = ½ x. C A B M

84. Remote Time A.Always B.Sometimes C.Never D.I don’t know

85. A. Always B. Sometimes C. Never D. I don’t know A square is ____________ a rhombus

86. A. Always B. Sometimes C. Never D. I don’t know The diagonals of a parallelogram ____________ bisect the angles of the parallelogram.

87. A. Always B. Sometimes C. Never D. I don’t know A quadrilateral with one pairs of sides congruent and one pair parallel is _________________ a parallelogram.

88. A. Always B. Sometimes C. Never D. I don’t know The diagonals of a rhombus are ___________ congruent.

89. A. Always B. Sometimes C. Never D. I don’t know A rectangle ______________ has consecutive sides congruent.

90. A. Always B. Sometimes C. Never D. I don’t know A rectangle ____________ has perpendicular diagonals.

91. A. Always B. Sometimes C. Never D. I don’t know The diagonals of a rhombus ___________ bisect each other.

92. A. Always B. Sometimes C. Never D. I don’t know The diagonals of a parallelogram are ____________ perpendicular bisectors of each other.

93. 5.5 Trapezoids Objectives Apply the definitions and learn the properties of a trapezoid and an isosceles trapezoid.

94. Trapezoid A quadrilateral with exactly one pair of parallel sides. A B C D Trap. ABCD How does this definition differ from that of a parallelogram?

95. Anatomy Of a Trapezoid R S TV Base The bases are the parallel sides 1 pair of base angles 2 nd pair of base angles

96. Anatomy Of a Trapezoid R S TV Leg The legs are the non-parallel sides

97. Isosceles Trapezoid A trapezoid with congruent legs. J KL M What do you think the definition is based on the diagram? What do you think would happen if I folded this figure in half?

98. Theorem The base angles of an isosceles trapezoid are congruent. E F G H Supplementary What is something I can conclude about 2 of the angles (other than congruency) based on the markings of the diagram?

99. The Median of a Trapezoid A segment that joins the midpoints of the legs. A B C D X Y Note: this applies to any trapezoid

100. Theorem The median of a trapezoid is parallel to the bases and its length is the average of the bases. B C D X Y A A B C D X Y How do we find an average of the bases ? Note: this applies to any trapezoid

101. White Board Practice B C D X Y A Complete 1. AD = 25, BC = 13, XY = ______ 19

102. White Board Practice B C D X Y A Complete 2. AX = 11, YD = 8, AB = _____, DC = ____ 2216

103. White Board Practice B C D X Y A Complete 3. AD = 29, XY = 24, BC =______ 19

104. White Board Practice B C D X Y A Complete 4. AD = 7y + 6, XY = 5y -3, BC = y – 5, y =____ 3.5

105. Test Review Know what properties each type of quadrilateral has –i.e. – all sides are congruent: square and rhombus –#1 – 10 on pg 187 Solving algebraic problems with a parallelogram –i.e. – finding the length of a side, angle, or diagonal Proving a quad is a parallelogram –Do you have enough information to say that the quad is a parallelogram –Knowing the 5 ways to prove

106. Solving problems using TH. 5-9 –I.e. #10 – 15 on pg. 180 Solving problems using Th. 5-11 and 5-15 –I.e. #1 – 4 on pg. 180 –I.ed # 14, 17 on page 187 Trapezoid Theorems –I.e. # 1 – 9 on pg. 192 –#11 pg 193