1. Unit 6
Quadrilaterals

2. Properties of Quadrilaterals
Lesson 6.1
Properties of Quadrilaterals

3. Lesson 6.1 Objectives Identify a figure to be a quadrilateral.
Use the sum of the interior angles of a quadrilateral. (G1.4.1)

4. Definition of a Quadrilateral
A quadrilateral is any four-sided figure with the following properties:
All sides must be line segments.
Each side must intersect only two other sides.
One at each of its endpoints, so that there are no:
Gaps that do not connect one side to another, or
Tails that extend beyond another side.

5. Example 6.1 Determine if the figure is a quadrilateral. Yes No No Yes
Too many intersecting segments No
Yes
No gaps
Yes No
Too many sides
No tails No No
No curves

6. Interior Angles
Recall that the interior angles of any figure are located in the interior and are formed by the sides of the figure itself.
Review: How many degrees does a straight line measure?
Review: What do you think the sum of the interior angles of a quadrilateral might be?
Review: What is the sum of the interior angles of any triangle?

7. Theorem 6.1: Interior Angles of a Quadrilateral Theorem
The sum of the measures of the interior angles of a quadrilateral is 360o. 1 2 3 4
m 1 +m 2 + m 3 + m 4 =
360o

8. Example 6.2
Find the missing angle.

9. Example 6.3
Find the x.

10. Lesson 6.1 Homework Lesson 6.1 – Properties of Quadrilaterals
Due Tomorrow

11. Lesson 6.2
Day 1:
Parallelograms

12. Lesson 6.2 Objectives Define a parallelogram
Define special parallelograms
Identify properties of parallelograms (G1.4.3)
Use properties of parallelograms to determine unknown quantities of the parallelogram (G1.4.4)

13. Definition of a Parallelogram
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

14. Theorem 6.2: Congruent Sides of a Parallelogram
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
The converse is also true!
Theorem 6.6

15. Theorem 6.3: Opposite Angles of a Parallelogram
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
The converse is also true!
Theorem 6.7

16. Example 6.4 x = 11 c – 5 = 20 m = 101 c = 25 y = 8 d + 15 = 68 d = 53
Find the missing variables in the parallelograms.     
x = 11
c – 5 = 20
m = 101
c = 25
y = 8
d + 15 = 68
d = 53

17. Theorem 6.4: Consecutive Angles of a Parallelogram
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
The converse is also true!
Theorem 6.8 Q P R S
m P + m S = 180o
m Q + m R = 180o
m P + m Q = 180o
m R + m S = 180o

18. Theorem 6.5: Diagonals of a Parallelogram
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Remember that means to cut into two congruent segments.
And again, the converse is also true!
Theorem 6.9

19. Example 6.5 Find the indicated measure in  HIJK HI GH KH HJ m KIH16
Theorem 6.2 GH 8
Theorem 6.5 KH 10 HJ
Theorem 6.5 & Seg Add Post
m KIH
28o
AIA Theorem
m JIH
96o
Theorem 6.4
m KJI
84o
Theorem 6.3

20. Theorem 6.10: Congruent Sides of a Parallelogram
If a quadrilateral has one pair of opposite sides that are both congruent and parallel, then it is a parallelogram.

21. Example 6.6 Yes! Yes! Yes! Yes! Yes! Yes!
Is there enough information to prove the quadrilaterals to be a parallelogram. If so, explain.
Yes!
Yes!
Yes!
Both pairs of opposite sides are congruent. (Theorem 6.6)
One pair of parallel and congruent sides.
(Theorem 6.10)
Both pairs of opposite angles are congruent. (Theorem 6.7)
Yes!
Yes!
Yes!
OR One pair of parallel and congruent sides. (Theorem 6.10)
Both pairs of opposite angles are congruent. (Theorem 6.7)
All consecutive angles are supplementary. (Theorem 6.8)
The diagonals bisect each other. (Theorem 6.9)
Both pairs of opposite sides are congruent. (Theorem 6.6)

22. Lesson 6.2a Homework
Lesson 6.2: Day 1 – Parallelograms
Due Tomorrow

23. Day 2: (Special) Parallelograms
Lesson 6.2
Day 2:
(Special) Parallelograms

24. Rhombus A rhombus is a parallelogram with four congruent sides.
The rhombus corollary states that a quadrilateral is a rhombus if and only if it has four congruent sides.

25. Theorem 6.11: Perpendicular Diagonals
A parallelogram is a rhombus if and only if its diagonals are perpendicular.

26. Theorem 6.12: Opposite Angle Bisector
A parallelogram is a rhombus iff each diagonal bisects a pair of opposite angles.

27. Rectangle A rectangle is a parallelogram with four congruent angles.
The rectangle corollary states that a quadrilateral is a rectangle iff it has four right angles.

28. Theorem 6.13: Four Congruent Diagonals
A parallelogram is a rectangle iff all four segments of the diagonals are congruent.

29. Square
A square is a parallelogram with four congruent sides and four congruent angles.

30. Square Corollary
A quadrilateral is a square iff its a rhombus and a rectangle.
So that means that all the properties of rhombuses and rectangles work for a square at the same time.

31. Example 6.7 Classify the parallelogram. Explain your reasoning.
Must be supplementary   
Rhombus
Rectangle
Square
Diagonals are perpendicular.
Theorem 6.11
Diagonals are congruent.
Theorem 6.13
Square Corollary

32. Lesson 6.2b Homework
Lesson 6.2: Day 2 – Parallelograms
Due Tomorrow

33. Lesson 6.3
Day 1:
Trapezoids

34. Lesson 6.3 Objectives Identify properties of a trapezoid. (G1.4.1)
Recognize an isosceles trapezoid. (G1.4.1)
Utilize the midsegment of a trapezoid to calculate other quantities from the trapezoid.
Identify a kite. (G1.4.1)

35. Trapezoid
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
The parallel sides are called the bases.
The nonparallel sides are called legs.
The angles formed by the bases are called the base angles.

36. Example 6.8 Find the indicated angle measure of the trapezoid. CIA CIA
Consecutive Interior Angles are supplementary!
CIA
CIA
Recall that a trapezoid has one set of parallel bases.

37. Example 6.9 Find x in the trapezoid. CIA CIA
Consecutive Interior Angles are supplementary!
Find x in the trapezoid.
CIA
CIA

38. Isosceles Trapezoid
If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.

39. Theorem 6.14: Bases Angles of a Trapezoid
If a trapezoid is isosceles, then each pair of base angles is congruent.
That means the top base angles are congruent.
The bottom base angles are congruent.
But they are not all congruent to each other!

40. Theorem 6.15: Base Angles of a Trapezoid Converse
If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid.

41. Theorem 6.16: Congruent Diagonals of a Trapezoid
A trapezoid is isosceles if and only if its diagonals are congruent.
Notice this is the entire diagonal itself.
Don’t worry about it being bisected cause it’s not!!

42. Example 6.10 Find the measures of the other three angles.
Supplementary
because of CIA 
127o
127o
83o  
97o
83o
Supplementary
because of CIA

43. Lesson 6.3a Homework
Lesson 6.3: Day 1 – Trapezoids
Due Tomorrow

44. Day 2: (More) Trapezoids and Kites
Lesson 6.3
Day 2:
(More) Trapezoids
and
Kites

45. Midsegment
The midsegment of a trapezoid is the segment that connects the midpoints of the legs of a trapezoid.

46. Theorem 6.17: Midsegment Theorem for Trapezoids
The midsegment of a trapezoid is
parallel to each base and
its length is one half the sum of the lengths of the bases.
It is the average of the base lengths! A B C D M N

47. Or essentially double the midsegment!
Example 6.11
Find the indicated length of the trapezoid. ? ? ?
Multiply both sides by 2.
Or essentially double the midsegment!

48. Kite
A kite is a quadrilateral that has two pairs of consecutive sides that are congruent, but opposite sides are not congruent.
It looks like the kite you got for your birthday when you were 5!
There are no sides that are parallel.

49. Theorem 6.18: Diagonals of a Kite
If a quadrilateral is a kite, then its diagonals are perpendicular.

50. Theorem 6.19: Opposite Angles of a Kite
If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.
The angles that are congruent are between the two different congruent sides.
You could call those the shoulder angles.
NOT

51. Example 6.12 Find the missing angle measures. But K  M
125o
64o
But K  M
125o 
88o
60 + K M = 360
60 + M M = 360
K = 88
M = 360
J = 360
2M = 250
M = 125
296 + J = 360
J = 64
K = 125

52. Example 6.13 Find the lengths of all the sides of the kite.
Round your answer to the nearest hundredth.
a2 + b2 = c2
= c2
a2 + b2 = c2
= c2
7.07
7.07
= c2
50 = c2
= c2
c = 7.07
169 = c2 13 13
c = 13
Use Pythagorean Theorem!
Cause the diagonals are perpendicular!!
a2 + b2 = c2

53. Lesson 6.3b Homework Lesson 6.3: Day 2 – (More) Trapezoids and Kites
Due Tomorrow

54. Perimeter and Area of Quadrilaterals
Lesson 6.4
Perimeter and Area of
Quadrilaterals

55. Lesson 6.4 Objectives
Find the perimeter of any type of quadrilateral. (G1.4.1)
Find the area of any type of quadrilateral. (G1.4.3)

56. Postulate 22: Area of a Square Postulate
The area of a square is the square of the length of its side. s

57. Theorem 6.20: Area of a Rectangle
The area of a rectangle is the product of a base and its corresponding height.
Corresponding height indicates a segment perpendicular to the base to the opposite side. h b

58. Example 6.14
Find the perimeter and area of the given quadrilateral.

59. Theorem 6.21: Area of a Parallelogram
The area of a parallelogram is the product of a base and its corresponding height.
Remember the height must be perpendicular to one of the bases.
The height will be given to you or you will need to find it.
To find it, use Pythagorean Theorem
a2 + b2 = c2 h b

60. Theorem 6.23: Area of a Trapezoid
The area of a trapezoid is one half the product of the height and the sum of the bases.
The height is the perpendicular segment between the bases of the trapezoid. b1 h b2

61. Theorem 6.24: Area of a Kite d1 d2
The area of a kite is one half the product of the lengths of the diagonals.
Hint: You must know the entire length of each diagonal before you begin multiplication. d1
Are you sensing a theme? d2

62. Theorem 6.25: Area of a Rhombus
The area of a rhombus is equal to one half the product of the lengths of the diagonals.
Again: You must know the entire length of each diagonal before you begin multiplication. d1
When all else fails, multiply the lengths that are perpendicular! d2

63. Example 6.15
Find the perimeter and area of the given quadrilateral.

64. Area Postulates Postulate 23: Area Congruence Postulate
If two polygons are congruent, then they have the same area.
Postulate 24: Area Addition Postulate
The area of a region is the sum of the areas of its nonoverlapping parts.

65. Example 6.16
Find the perimeter and area of the given figure. Assume all corners form a right angle.

66. Side was multiplied by 3, so the area gets multiplied by 32 = 9.
Example 6.17
Square A has a side length of 2 ft. If you double the length of the side, what happens to the area of the square?
A The area stays the same. B The area doubles. C The area quadruples. D Not enough information.
Since the formula is A = s2, any multiplied change to a side will change the area by that multiple “squared”!
Square B has a side length of 4 ft. If you triple the length of the side, what happens to the area of the square?
A The area stays the same. B The area triples. C The area quadruples. D The area increases by a factor of nine.
Side was multiplied by 3, so the area gets multiplied by 32 = 9.

67. Example 6.18
A rectangle has a base of 5 ft and a height of 4 ft. If you double the base and leave the height the same, what happens to the area of the rectangle?
A The area stays the same. B The area doubles. C The area quadruples. D Not enough information.
Since the formula is A = bh, and the change only occurred to one side, the area will be multiplied by that single multiplied change.
A rectangle has a base of 8 ft and a height of 6 ft. If you triple the base and leave the height the same, what happens to the area of the rectangle?
A The area stays the same. B The area doubles. C The area triples. D The area quadruples.
A1 = (8 ft)(6 ft) = 48 ft2 A2 = (3•8 ft)(6 ft) A2 = (24 ft)(6 ft) = 144 ft2 = (3•48) ft2

68. Lesson 6.4 Homework Lesson 6.4 – Perimeter and Area of Quadrilaterals
Due Tomorrow

69. Quadrilateral Hierarchy
Lesson 6.5
Quadrilateral Hierarchy

70. Lesson 6.5 Objectives Create a hierarchy of polygons.
Identify special quadrilaterals based on given information. (G1.4.3)

71. Polygon Hierarchy Polygons Pentagons Triangles Quadrilaterals
Parallelogram
Trapezoid
Kite
Rhombus
Rectangle
Isosceles Trapezoid
Square

72. How to Read the Hierarchy
NEVER
How to Read the Hierarchy
Polygons
Triangles
Quadrilaterals
Pentagons
Rhombus
Rectangle
Trapezoid
Parallelogram
Kite
Square
Isosceles Trapezoid
ALWAYS
SOMETIMES
But a parallelogram is sometimes a rhombus and sometimes a square.
So that means that a square is always a rhombus, a parallelogram, a quadrilateral, and a polygon.
However, a parallelogram is never a trapezoid or a kite.

73. Using the Hierarchy
When asked to identify a quadrilateral as specific as possible,
Identify which branch of the “family tree” it belongs to
Parallelogram
Trapezoid, or
Kite
Then start at the bottom and work your way up that branch of the “family tree.”
For instance, start at square and test if the properties of a square fit.
If not all properties fit, then go up to the next level of the hierarchy until you find where all the properties fit.
Remember that a quadrilateral must fit all the properties of its “ancestors.”
So a square must have the same properties of:
A rhombus, and
A rectangle, and
A parallelogram
Because a rhombuses and rectangles are also parallelograms

74. Example 6.19 State the most specific name for each figure Rectangle
Parallelogram
Rhombus
Square
Parallelogram
Isosceles Trapezoid

75. Example 6.20 Fill in the blanks with sometimes, always, or never.
A parallelogram is ____________ a square.
sometimes
A trapezoid is ____________ an isosceles trapezoid.
A rectangle is _____________ a kite.
never
A square is ____________ a rectangle.
always
A rectangle is ____________ a square.
A rhombus is ____________ a rectangle.
A rhombus is ___________ a sqaure.
A kite is ____________ a quadrilateral.

76. Example 6.21
Tell if the condition given is necessary, sufficient, both, or neither.
To be a rhombus: Condition: Be a quadrilateral with 4 equal sides.
Both
To be a kite: Condition: Be a figure with 4 sides.
Necessary
To be an isosceles trapezoid: Condition: Be a figure one pair of opposite sides parallel.
To be a parallelogram: Condition: Be a figure with two pairs of opposite sides parallel.
To be a rectangle: Condition: Be a parallelogram with four right angles.

77. Homework 6.5
Lesson 6.5 – Quadrilateral Hierarchy
Due Tomorrow