1. Chapter 6 Notes

2. 6.1 – Polygons

3. Poly  Many
A polygon is a plane figure that meets the following conditions:
1) It is formed by three or more segments called sides such that no two sides with a common endpoint are collinear.
2) Each side intersects exactly two other sides, one at each endpoint.

4. Names of polygons
# of sides Name
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
8 Octagon
10 decagon
12 dodecagon
n n-gon
When naming polygons, you list the vertices in order! P R E T L B
Hexagon PRTBLE Or
Hexagon TBLEPR
Or other names

5. Convex  A polygon such that no line containing a side of a polygon contains a point in the interior of the polygon.

6. Equilateral  Sides are the same.
Equiangular  Angles are the same.
Regular  Both

7. Diagonal – Segment that joins two nonconsecutive vertices
Is it a polygon? If so, name it and say if it is convex or concave.
Diagonal – Segment that joins two nonconsecutive vertices

8. Interior angles of a Quadrilateral
The sum of the measures of the interior angles of a quadrilateral is 360o.
Solve for x xo xo
2xo
2xo
x+20o
x+20o
100o
100o

9. (3x + 2)o
(2x – 10 )o
(x + 5)o
(2x – 7)o

10. 6.2 – Properties of Parallelograms

11. Definition of a parallelogram  Both pairs of opposite sides are parallel.C M

12. A B D C M
Find all information

13. x + 24 z 65 y
4x – 12

14. zo 7y wo
9q + 4
3y + 28 xo
30o
30o
write

15. Proving Theorem 6.2 A B D C

16. A B D C E F

17. B X C 1 F E Y 2 A Z D

18. 6.3 – Proving Quadrilaterals are Parallelograms

19. Opposite sides are parallel, then it’s a ||-gram by def.
THRM 6.6 If both pairs of opposite sides of a quad are congruent, then the quadrilateral is a parallelogram.
THRM 6.8, If an angle of a quad is supplementary to both its consecutive angles, then the quadrilateral is a ||- gram T Q S R 1 4 2 3
THRM 6.7 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a ||-gram
THRM 6.9 If the diagonals of a quad bisect each other, then the quadrilateral is a ||-gram

20. Let’s discuss how to prove two of these theorems
THRM 6.6 If both pairs of opposite sides of a quad are congruent, then the quadrilateral is a parallelogram. T Q S R 1 4 2 3 M
THRM 6.9 If the diagonals of a quad bisect each other, then the quadrilateral is a ||-gram

21. Yes parallelogram, not a parallelogram, and why?
THRM 6.10, If one pair of opposite sides are both CONGRUENT and PARALLEL, then the quadrilateral is a ||-gram T Q S R 1 4 2 3 M
Yes parallelogram, not a parallelogram, and why?

22. Prove that the following coordinates make a parallelogram by the given theorems\definitions using slope formula and distance formula.
Opposite sides are parallel, then it’s a ||-gram by def.
One pair of opposite sides are both CONGRUENT and PARALLEL.
The diagonals of a quad bisect each other
(-1, 3)
(3, 2)
(-4, -1)
(0, -2) A B D C

23. Do these points make a parallelogram?
(0,2) (-3,1) (-2, 3) (1,4)
Do these points make a parallelogram?
(-1,-3) (-2, 1) (2, 2) (1, -3)

24. 6.4 – Rhombuses, Rectangles, and Squares

25. Rectangle – Quad with 4 rt angles.
Rhombus – Quad with 4 congruent sides
Square – 4 rt angles AND 4 congruent sides, It’s a rectangle AND a rhombus!!
Why are these parallelograms?

26. World O’ Parallelograms
Answer with always, sometimes, or never
A Rhombus is a Square Always Sometimes Never
A Square is a Parallelogram Always Sometimes Never

27. Corollaries
Rhombus Corollary A quadrilateral is a rhombus iff it has four congruent sides.
Rectangle Corollary A quadrilateral is a rectangle iff it has four right angles.
Square Corollary  A quadrilateral is a square iff it is a rhombus and a rectangle.

28. Thrm 6.11  A ||-gram is a rhombus iff its diagonals are perpendicular.
Thrm 6.12  A ||-gram is a rhombus iff each diagonal bisects a pair of opposite angles. A B D C

29. Thrm 6.13  A ||-gram is a rectangle iff its diagonals are congruent.B D C
Prove this theorem.
Stuwork

30. Which shape could it be?
What can be true about it?

31. Given the figure is a rectangle,
Solve for x and y.
Given the figure is a rectangle,
Solve for x and y.
2y + 35
70o xo
2y + 16
(2x + 5)o
4y – 10
5y – 10
Stuwork

32. Given the figure is a square,
Solve for x and y.
Given the figure is a square,
Solve for x and y.
55o A B xo 10 yo 5 y D C
AC = 4x – 10
BD = 2x + 2
Stuwork

33. A ||-gram is a rectangle iff its diagonals are congruent
A ||-gram is a rectangle iff its diagonals are congruent. We’ll now prove this using coordinate proof
( , )
( , ) A B D C
( , )
( , )
Stuwork

34. Well do some coordinate proof stuff with rhombus, rectangles, and squares added on.

35. 6.5 – Trapezoids and Kites

36. A quadrilateral with EXACTLY one pair of parallel sides is called a TRAPEZOID. The parallel sides are called BASES. The other sides are LEGS
Trapezoids have two pairs of base angles.
ISOSCELES TRAPEZOID – LEGS are CONGRUENT!
KITE – A quadrilateral with two pairs of consecutive sides, BUT opposite sides aren’t congruent.

37. Theorem 6.14  Base angles of an isosceles trapezoid are congruent.
ONLY TRUE FOR ISOS TRAPEZOID, NOT REGULAR TRAPEZOID! X Y A B Z
Congruent, opp sides ||-gram congruent.
Transitive to work all sides congruent.
Corresponding, base angles thrm, transitive.
Same side interior angles, measure, supp, subtraction.

38. Theorem 6.15 If a trapezoid has one pair of congruent base angles, then it’s an isosceles trapezoid. A B C D
Theorem 6.16 A trapezoid is isosceles iff its diagonals are congruent. A B C D

39. Just like in a triangle, the midsegment will go through the midpoint of the legs.B C D b1 b2
midsegment E F

40. A B C D 10 18 x E F A B C D 12 y 15 E F

41. A B C D z 14 11 E F A B C D 14
2x+2
5y+10
7y-10

42. x 13 y 19 z A B C D R AD = 15 AR = 6 BC = _____ BR = __
Write in A B C D R
AD = 15 AR = 6
BC = _____ BR = __
RC = ____ RD = __

44. 55 22 y x 44 45 z

45. C
Theorem 6.18
If a quadrilateral is a kite, then its diagonals are perpendicular B D A
Theorem 6.18
If a quadrilateral is a kite, then EXACTLY one pair of opposite angles are congruent. C B D A

46. A B D C x 12 5 A B D C
60o
140o zo yo

47. A B D C 25 24 x A B D C zo
100o
70o yo

48. Go Over HW
Verify Quadrilateral while HW checked
Properties of shapes of Parallelogram, Rhombus, Rectangle, Square
Discuss what’s on quiz

49. 6.6 – Special Quadrilaterals

50. Quad-Tree-Laterals

51. For which shape(s) is/are the following always true?
Make a table with the following shapes as the heading:
Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Isosceles Trapezoid
Kite

52. 6.7 – Areas of Triangles and Quadrilaterals

53. Postulate: The area of a square is the square of the length of its side  A = s2
Area congruence Postulate  If two figures are congruent, they have the same area.

54. Area Addition Postulate  The area of a region is the sum of the areas of the non overlapping part. Just means you can cut up the parts and add them together.

55. In a parallelogram, a side can be considered the base
In a parallelogram, a side can be considered the base. The line perpendicular to the base and going to the other side is the altitude. The length of the altitude is called the height. (Altitude  segment, height  number) a h b b
Area of a rectangle equals the product of the base and height. (A = bh)

56. Area of Parallelogram
A = bh h b

57. b1 h b2

58. d2 d1
Kite works the same way.

59. 22 22 5 10 10 4 10 15 4 8 5 10
Write bottom two

60. The sides of a rectangle are x and x – 14. The area is 120
The sides of a rectangle are x and x – 14. The area is Find the value of x.
x – 14 x

61. Area is 40 ft2. Find h
Area is 80 ft2. Find x 10 x h x 16

62. 12 17 8 8 12 6

63. 4 5 11 4 14 6 14

64. Find Area
Write bottom two

65. Find Area
Write bottom two