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
Trapezoids
and
Kites

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

44. 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

45. 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!

46. 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.

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

48. 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

49. 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

50. 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

51. Lesson 6.3 Homework
Lesson 6.3 – Trapezoids and Kites
Due Tomorrow

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

53. 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)

54. Postulate 22: Area of a Square Postulate
The area of a square is the square of the length of its side.
A = s2 s

55. 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.
A = bh h b

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

57. 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
A = bh h b

58. 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.
A = ½ (b1+b2) h b1 h b2

59. 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.
A = ½ d1d2 d1 d2

60. 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.
A = ½ d1d2 d1 d2

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

62. 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.

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

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

65. Special Quadrilaterals
Lesson 6.5
Special Quadrilaterals

66. Lesson 6.6 Objectives Create a hierarchy of polygons
Identify special quadrilaterals based on limited information

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

68. 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.

69. Using the Hierarchy
Remember that a square must fit all the properties of its “ancestors.”
That means the properties of a rhombus, rectangle, parallelogram, quadrilateral, and polygon must all be true!
So when asked to identify a figure as specific as possible, test the properties working your way down the hierarchy.
As soon as you find a figure that doesn’t work any more you should be able to identify the specific name of that figure.

70. Homework 6.6 HW p
8-35, 55-65
Due Tomorrow
Test Friday
March 26