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 ( )
What do you think the definition is based on the diagram?
If a quadrilateral is a parallelogram, then both pairs of opposite sides are parallel.
ABCD A B C D
Partners: What do we know about the angles of a parallelogram b/c it has parallel sides?
What do we know a bout a quadrilateral besides the fact that it has four sides and four angles? What do we know about a parallelogram because it has parallel sides?

7. Naming a Parallelogram
Use the symbol for parallelogram and name using the 4 vertices in order either clockwise or counter clockwise.
ABCD A B C D
What do we know a bout a quadrilateral besides the fact that it has four sides and four angles? What do we know about a parallelogram because it has parallel sides?

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

9. Theorem Opposite sides of a parallelogram are congruent. A B D
Do the proof. D
What would be our plan for solving this theorem? C

10. Theorem Opposite angles of a parallelogram are congruent.
What did we discuss at the beginning of the lesson about s-s int. angles? A B
Do the proof. Note that it is only possible to do the angles one pair at a time. D C

11. Theorem The diagonals of a parallelogram bisect each other. A B D
Do the proof. D
What is another name for AC and BD? C

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

13. Remote Time
True or False

14. True or False
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 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. White Board Groups
Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. 6
x = 80
y = 100
a = 6
b = 9 R S yº xº 9 b
80º T U a

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

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

29. 5-2:Ways to Prove that Quadrilaterals are Parallelograms
Objectives
Learn about ways to prove a quadrilateral is a parallelogram
Most of today's statements are the converses of yesterdays. The properties of a parallelogram can be used to prove that 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. A B C D
The definition is a biconditional, so it can be used either way.
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
Do the proof. Emphasize that no properties of parallelograms can be used in proving this, because it won’t be a parallelogram until the proof is complete. D C

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
Ditto. D C

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. A B
Go through an outline of this proof, but do not do it formally. This proof is unusual because the construction of an auxiliary line will not help. This proof cannot be done with congruent triangles. D C

36. Theorem
If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Show that the diagonals bisect each other. A B
A more conventional proof. X D C

37. Five ways to prove a Quadrilateral is a Parallelogram
Show that both pairs of opposite sides parallel
Show that both pairs of opposite sides congruent
Show that one pair of opposite sides are both congruent and parallel
Show that both pairs of opposite angles congruent
Show that diagonals that bisect each other
While all of these are useful, and most proofs offer you the choice of method, the third one (Th. 5-5) is the best to use, if possible, because it only involves one pair of sides.

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

39. If the measure of two angles of a quadrilateral are equal, then the quadrilateral is ____________ a parallelogram
Sometimes
Always
Never
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.
Demo: 6 volunteers
How do we measure the distance from a point to a line?
This is another way of defining parallel lines. m n
What does equidistant mean?

46. Demo: Popsicle Sticks and lined paper
Theorem
If three parallel lines cut off congruent segments on one transversal, then they do so on any transversal. A D B
A good memory device for this theorem is to call it the “Knife Theorem” because it is like a chefs knife, always cutting uniformly, but cutting different things into different shapes. E C F

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 1 2 3 A 3
A good device for this theorem is the “1-2-3 Theorem” because all three sides of the triangle are involved. X Y 1 2 B C

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. 1 2 3 A
If BC is 12 then XY =? 2
This is the converse to the “1-2-3 Theorem”. Its proof is interesting, especially the part about being half the length. X Y 1 3 B C

49. White Board Practice AB BC AC ST RT RS 12 14 18 15 22 10 9 7.8
Given: R, S, and T are midpoint of the sides of  ABC AB BC AC ST RT RS 12 14 18 15 22 10 9
7.8 B R S C A T

50. White Board Practice AB BC AC ST RT RS 12 14 18 6 7 9 20 15 22 10 7.5
Given: R, S, and T are midpoint of the sides of  ABC AB BC AC ST RT RS 12 14 18 6 7 9 20 15 22 10
7.5 11
15.6 5
7.8 B R S C A T

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

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

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

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

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

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

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

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

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

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

61. QUIZ REVIEW Know the properties of a parallelogram
Know the 5 ways to prove a quad is a parallelogram
“Show that…..
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..
Both pairs of opposite sides are parallel
From theorems…
Both pairs of opposite sides are congruent
Both pairs of opposite angles are congruent
The diagonals of a parallelogram bisect each other

64. QUADRILATERALS
parallelogram
Rhombus
Rectangle
Square

65. Rectangle By definition, it is a quadrilateral with four right angles.V
While basic, all these definitions also include all of the properties of a parallelogram. Focus not only on what a rectangle has in common with a parallelogram, but what distinguishes them as well. S T

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

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

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

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. K X J L
Prove this. Show the difference between a kite and a rhombus.
What does the definition of perpendicular lines tell us? M

71. Theorem Each diagonal of a rhombus bisects the opposite angles. K X J
Prove this if there is time. J L M

72. Theorem
If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle. R V
WHY?????????
This indicates the minimum burden of proof to make a parallelogram into a rectangle. S T

73. Theorem
If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. B C
WHY?
This indicates the minimum burden of proof to make a parallelogram into a rhombus. A D

74. X M Z Y

75. Theorem
The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. X
Be sure to prove this. Do an extensive analysis as it involves some isosceles triangles as well. M Z Y

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

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 A B E
62º D C

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 = M N
29º 12 L P O

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 M N
29º 12 L P O

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

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 . A M B C

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

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 . A M B C

84. Remote Time
Always
Sometimes
Never
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. B C
Trap. ABCD C
There is no symbol for trapezoid. Be sure to note that the definitions of a parallelogram and a trapezoid make them mutually exclusive.
How does this definition differ from that of a parallelogram? A D

95. Anatomy Of a Trapezoid The bases are the parallel sides Base R S V T
This slide allows you to define the bases, legs and base angles of a trapezoid. Animate the parts and use the smart pens to write the labels at the appropriate time.
1 pair of base angles
2nd pair of base angles V T
Base

96. Anatomy Of a Trapezoid The legs are the non-parallel sides R S Leg Leg
This slide allows you to define the bases, legs and base angles of a trapezoid. Animate the parts and use the smart pens to write the labels at the appropriate time.
Leg
Leg V T

97. Isosceles Trapezoid A trapezoid with congruent legs. J 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?
Point out that the bases cannot be congruent and the legs cannot be parallel. K L

98. Theorem The base angles of an isosceles trapezoid are congruent. F G E
Supplementary
There are two pairs of base angles, and in addition to being congruent, they are supplementary to the other pair.
Supplementary
What is something I can conclude about 2 of the angles (other than congruency) based on the markings of the diagram? E H

99. The Median of a Trapezoid
A segment that joins the midpoints of the legs. B C
While the median of a triangle connects the vertex to the midpoint of the opposite side, a trapezoid has no vertex opposite each side, it has an equal number of sides. X Y
Note: this applies to any trapezoid A D

100. Theorem
The median of a trapezoid is parallel to the bases and its length is the average of the bases.
Note: this applies to any trapezoid B B C C
Both of these things appear to be true about a median. Focus on that and try to get them to see that its length is the average. The sketch shows the relationship in a movable diagram. X X Y Y
How do we find an average of the bases ? A A D D

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

102. White Board Practice 22 16 Complete
2. AX = 11, YD = 8, AB = _____, DC = ____ 22 16 B C X Y A D

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

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

105. Test Review Solving algebraic problems with a parallelogram
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 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