1. Chapter 6
Quadrilaterals

2. Section 6.1
Polygons

3. Polygon A polygon is formed by three or more segments called sides
No two sides with a common endpoint are collinear.
Each side intersects exactly two other sides, one at each endpoint.
Each endpoint of a side is a vertex of the polygon.
Polygons are named by listing the vertices consecutively.

4. Identifying polygons
State whether the figure is a polygon. If not, explain why.

5. Polygons are classified by the number of sides they have
TYPE OF POLYGON 3 4 5 6 7
NUMBER OF SIDES
TYPE OF POLYGON 8 9 10 12
N-gon
octagon
triangle
nonagon
quadrilateral
pentagon
decagon
dodecagon
hexagon
heptagon
N-gon

6. Two Types of Polygons:
Convex: If a line was extended from the sides of a polygon, it will NOT go through the interior of the polygon.
Example:

7. 2. Concave: If a line was extended from the sides of a polygon, it WILL go through the interior of the polygon.
Example:

9. Regular Polygon
A polygon is regular if it is equilateral and equiangular
A polygon is equilateral if all of its sides are congruent
A polygon is equiangular if all of its interior angles are congruent

11. Diagonal
A segment that joins two nonconsecutive vertices.

13. Interior Angles of a Quadrilateral Theorem
The sum of the measures of the interior angles of a quadrilateral is 360° 1 4 2 3

16. Properties of Parallelograms
Section 6.2
Properties of Parallelograms

17. Parallelogram
A quadrilateral with both pairs of opposite sides parallel

18. Theorem 6.2
Opposite sides of a parallelogram are congruent.

19. Theorem 6.3
Opposite angles of a parallelogram are congruent

20. Theorem 6.4 Consecutive angles of a parallelogram are supplementary. 21 2 3 4

21. Theorem 6.5
Diagonals of a parallelogram bisect each other.

25. Proving Quadrilaterals are Parallelograms
Section 6.3
Proving Quadrilaterals are Parallelograms

26. Theorem 6.6 To prove a quadrilateral is a parallelogram:
Both pairs of opposite sides are congruent

27. Theorem 6.7 To prove a quadrilateral is a parallelogram:
Both pairs of opposite angles are congruent.

28. Theorem 6.8 To prove a quadrilateral is a parallelogram:
An angle is supplementary to both of its consecutive angles. 1 2 3 4

29. Theorem 6.9 To prove a quadrilateral is a parallelogram:
Diagonals bisect each other.

30. Theorem 6.10 To prove a quadrilateral is a parallelogram:
One pair of opposite sides are congruent and parallel.
>
>

33. Types of parallelograms
Section 6.4
Types of
parallelograms

34. Rhombus
Parallelogram with four congruent sides.

35. Properties of a rhombus
Diagonals of a rhombus are perpendicular.

36. Properties of a rhombus
Each Diagonal of a rhombus bisects a pair of opposite angles.

37. Rectangle
Parallelogram with four right angles.

38. Properties of a rectangle
Diagonals of a rectangle are congruent.

39. Square
Parallelogram with four congruent sides and four congruent angles.
Both a rhombus and rectangle.

40. Properties of a square
Diagonals of a square are perpendicular.

41. Properties of a square
Each diagonal of a square bisects a pair of opposite angles.
45°
45°
45°
45°
45°
45°
45°
45°

42. Properties of a square
Diagonals of a square are congruent.

43. 3-Way Tie
Rectangle
Rhombus
Square

45. Section 6.5
Trapezoids and Kites

46. Trapezoid
Quadrilateral with exactly one pair of parallel sides.
Parallel sides are the bases.
Two pairs of base angles.
Nonparallel sides are the legs.
Base
>
Leg
Leg
>
Base

47. Isosceles Trapezoid
Legs of a trapezoid are congruent.

48. Theorem 6.14 > Base angles of an isosceles trapezoid are congruent.

49. Theorem 6.15
If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid.
> A B
> D C
ABCD is an isosceles trapezoid

50. Theorem 6.16
Diagonals of an isosceles trapezoid are congruent.
>
> A B C D
ABCD is isosceles if and only if

51. Examples on Board

52. Midsegment of a trapezoid
Segment that connects the midpoints of its legs.
Midsegment

53. Midsegment Theorem for trapezoids
Midsegment is parallel to each base and its length is one half the sum of the lengths of the bases. A B C D M N
MN= (AD+BC)

54. Examples on Board

55. Kite
Quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

56. Theorem 6.18
Diagonals of a kite are perpendicular. A B C D

57. Theorem 6.19
In a kite, exactly one pair of opposite angles are congruent. A B C D

58. Examples on Board

59. Pythagorean Theorem c a b

60. Special Quadrilaterals
Section 6.6
Special Quadrilaterals

61. Properties of Quadrilaterals
Property
Rectangle
Rhombus
Square
Trapezoid
Kite
Both pairs of opposite sides are congruent
Diagonals are congruent
Diagonals are perpendicular
Diagonals bisect one another
Consecutive angles are supplementary
Both pairs of opposite angles are congruent X X X X X X X X X X X X X X X X X X X X X

62. Properties of Quadrilaterals
Quadrilateral ABCD has at least one pair of opposite sides congruent. What kinds of quadrilaterals meet this condition?
PARALLELOGRAM
RECTANGLE
ISOSCELES
TRAPEZOID
SQUARE
RHOMBUS

63. Areas of Triangles and Quadrilaterals
Section 6.7
Areas of Triangles and Quadrilaterals

64. Area Congruence Postulate
If two polygons are congruent, then they have the same area.

65. Area Addition Postulate
The area of a region is the sum of the areas of its non-overlapping parts.

66. Area Formulas
TRIANGLE
RECTANGLE
SQUARE
PARALLELOGRAM
A=bh
A=lw

67. Area Formulas
RHOMBUS
KITE

68. Area Formulas
TRAPEZOID h