1. 6.1 Polygons
Geometry

2. Objectives/DFA/HW Objectives: DFA: HW:
You will solve problems using the interior & exterior angle-sum theorems.
DFA:
pp #16 & #30
HW:
pp (2-44 even)

3. What is polygon? Formed by three or more segments (sides).
Each side intersects exactly two other sides, one at each endpoint.
Has vertex/vertices.

4. Polygons are named by the number of sides they have. Fill in the blank.
Type of polygon 3
Triangle 4 5 6 7 8
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon

5. Concave vs. Convex
Convex: if no line that contains a side of the polygon contains a point in the interior of the polygon.
Concave: if a polygon is not convex.
interior

6. Example
Identify the polygon and state whether it is convex or concave.
Convex polygon
Concave polygon

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

8. Decide whether the polygon is regular.) )) ) ) ) ) )) ) )

9. A Diagonal of a polygon is a segment that joins two nonconsecutive vertices.
Have students find more diagonals
diagonals

10. Interior Angles of a Quadrilateral Theorem
The sum of the measures of the interior angles of a quadrilateral is 360°. B
m<A + m<B + m<C + m<D = 360° C A D

11. Example Find m<Q and m<R. x + 2x + 70° + 80° = 360°
m< Q = x
m< Q = 70 °
m<R = 2x
m<R = 2(70°)
m<R = 140 ° S

12. Find m<A C
65° D
55°
123° B A

13. Use the information in the diagram to solve for j.
60°
150°
3j °

14. Theorem 6-1 – Polygon Angle-Sum Theorem
The sum of the measures of the interior angles of an n-gon is (n-2)180.
Ex. What is the sum of the interior angle measures of a heptagon?

15. Theorem 6-2 Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles of polygon, one at each vertex is 360o.
For the petagon
m<1+m<2+m<3+m<4+m<5=360
Ex. What is the measure of each
angle of an octagon.

16. 6.2 Properties of Parallelograms
Geometry
Spring 2014

17. Objective/DFA/HW Objectives: DFA: HW:
You will use properties (angles & sides) of parallelograms & relationships among diagonals to solve problems relating to parallelograms.
DFA:
pp.364 #16 & #22
HW:
pp (2-40 even)

18. Theorems
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
If a quadrilateral is a parallelogram, then its opposite angles are congruent. Q R S P

19. Theorems
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
m<P + m<Q = 180°
m<Q + m<R = 180°
m<R + m<S = 180°
m<S + m<P = 180° Q R S P

20. Using Properties of Parallelograms
PQRS is a parallelogram. Find the angle measure.
m< R
m< Q
70 ° Q R
70 ° + m < Q = 180 °
m< Q = 110 °
70° P S

21. Using Algebra with Parallelograms
PQRS is a parallelogram. Find the value of h. P Q 3h
120° S R

22. Theorems
If a quadrilateral is a parallelogram, then its diagonals bisect each other. R Q M P S

23. Using properties of parallelograms
FGHJ is a parallelogram. Find the unknown length. JH JK 5 F 5 3 G 3 K J H

24. Examples
Use the diagram of parallelogram JKLM. Complete the statement. LM K L NK
<KJM N
<LMJ NL MJ J M

25. Find the measure in parallelogram LMNQ.LP LQ QP
m<LMN
m<NQL
m<MNQ
m<LMQ 18 8 L M 9
110° 10 10 9 P
70° 8
32°
70 ° Q N 18
110 °
32 °

26. Find X, Y, & the diagonals X
2x-8
Y+10
Y+2

27. Theorem 6.7
If 3 (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

28. 6.3 Proving Quadrilaterals are Parallelograms
Geometry
Spring 2014

29. Objective/DFA/HW Objectives: DFA: HW:
You will determine whether a quadrialteral is a parallelogram.
DFA:
pp.372 #12
HW:
pp (2-28 even, all)

30. Review

31. Using properties of parallelograms.
Method 1
Use the slope formula to show that opposite sides have the same slope, so they are parallel.
Method 2
Use the distance formula to show that the opposite sides have the same length.
Method 3
Use both slope and distance formula to show one pair of opposite side is congruent and parallel.

32. Let’s apply~
Show that A(2,0), B(3,4), C(-2,6), and D(-3,2) are the vertices of parallelogram by using method 1.

33. Show that the quadrilateral with vertices A(-3,0), B(-2,-4), C(-7, -6) and D(-8, -2) is a parallelogram using method 2.

34. Show that the quadrilateral with vertices A(-1, -2), B(5,3), C(6,6), and D(0,7) is a parallelogram using method 3.

35. Proving quadrilaterals are parallelograms
Show that both pairs of opposite sides are parallel.
Show that both pairs of opposite sides are congruent.
Show that both pairs of opposite angles are congruent.
Show that one angle is supplementary to both consecutive angles.

36. .. continued.. Show that the diagonals bisect each other
Show that one pair of opposite sides are congruent and parallel.

37. Show that the quadrilateral with vertices A(-1, -2), B(5,3), C(6,6), and D(0,7) is a parallelogram using method 3.

38. Example 4 – p.341
Show that A(2,-1), B(1,3), C(6,5), and D(7,1) are the vertices of a parallelogram.

39. Assignments In class: pp. 342-343 # 1-8 all
Homework: pp #10-18 even, 26, 37

40. 6.4 Rhombuses, Rectangles, and Squares
Geometry
Spring 2014

41. Objective/DFA/HW Objectives: DFA: HW:
You will determine whether a parallelogram is a rhombus, rectangle, or a square & you will solve problems using properties of special parallelograms.
DFA:
pp.379 #12
HW:
pp (1-27all)

42. Review Find the value of the variables. p + 50° + (2p – 14)° = 180°
52°
(2p-14)°
50°
68°
p + 50° + (2p – 14)° = 180°
p + 2p + 50° - 14° = 180°
3p ° = 180°
3p = 144 °
p = 48 °
52° + 68° + h = 180°
120° + h = 180 °
h = 60°

43. Special Parallelograms
Rhombus
A rhombus is a parallelogram with four congruent sides.

44. Special Parallelograms
Rectangle
A rectangle is a parallelogram with four right angles.

45. Special Parallelogram
Square
A square is a parallelogram with four congruent sides and four right angles.

46. Corollaries Rhombus corollary Rectangle corollary Square corollary
A quadrilateral is a rhombus if and only if it has four congruent sides.
Rectangle corollary
A quadrilateral is a rectangle if and only if it has four right angles.
Square corollary
A quadrilateral is a square if and only if it is a rhombus and a rectangle.

47. Example PQRS is a rhombus. What is the value of b? 2b + 3 = 5b – 6

48. Review In rectangle ABCD, if AB = 7f – 3 and CD = 4f + 9, then f = ___1 2 3 4 5
7f – 3 = 4f + 9
3f – 3 = 9
3f = 12
f = 4

49. Example PQRS is a rhombus. What is the value of b? 3b + 12 = 5b – 6

50. Theorems for rhombus
A parallelogram is a rhombus if and only if its diagonals are perpendicular.
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. L

51. Theorem of rectangle
A parallelogram is a rectangle if and only if its diagonals are congruent. A B D C

52. Match the properties of a quadrilateral
The diagonals are congruent
Both pairs of opposite sides are congruent
Both pairs of opposite sides are parallel
All angles are congruent
All sides are congruent
Diagonals bisect the angles
Parallelogram
Rectangle
Rhombus
Square
B,D
A,B,C,D
A,B,C,D
B,D
C,D C

53. 6.5 Trapezoid and Kites
Geometry
Spring 2014

54. • Objective
You will determine whether a parallelogram is a kite or a trapezoid.
• DFA –
P.387 # 18
o HW –
p (2-34 even)

55. Let’s define Trapezoid
base A B
>
leg
leg
> D C
base
<D AND <C ARE ONE PAIR OF BASE ANGLES.
When the legs of a trapezoid are congruent,
then the trapezoid is an isosceles trapezoid.

56. Isosceles Trapezoid
If a trapezoid is isosceles, then each pair of base angles is congruent. A B D C

57. PQRS is an isosceles trapezoid. Find m<P, m<Q, and m<R.
>
50° P
> Q

58. Isosceles Trapezoid
A trapezoid is isosceles if and only if its diagonals are congruent. A B D C

59. Midsegment Theorem for Trapezoid
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’s the average of the lengths of the bases. B C M N A D

60. Examples The midsegment of the trapezoid is RT. Find the value of x.7 R x T
x = ½ (7 + 14)
x = ½ (21)
x = 21/2 14

61. Examples The midsegment of the trapezoid is ST. Find the value of x.8 S 11 T
11 = ½ (8 + x)
22 = 8 + x
14 = x x

62. Review
In a rectangle ABCD, if AB = 7x – 3, and CD = 4x + 9, then x = ___
A) 1
B) 2
C) 3
D) 4
E) 5
7x – 3 = 4x + 9
-4x x
3x – 3 =
3x = 12
x = 4

63. Kite
A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

64. Theorems about Kites
If a quadrilateral is a kite, then its diagonals are perpendicular
If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. B A C L D

65. Example Find m<G and m<J. Since m<G = m<J,H
132°
60° K G

66. Example
Find the side length. J 12 H K 12 14 12 G

67. 6.6 Special Quadrilaterals
Geometry
Spring 2014

68. Summarizing Properties of Quadrilaterals
Kite
Parallelogram
Trapezoid
Isosceles Trapezoid
Rhombus
Rectangle
Square

69. Identifying Quadrilaterals
Quadrilateral ABCD has at least one pair of opposite sides congruent. What kinds of quadrilaterals meet this condition?
There are many solutions to this… please explore many possibilities with visuals.

70. Copy the chart. Put an X in the box if the shape
always has the given property.
Property
Parallelogram
Rectangle
Rhombus
Square
Kite
Trapezoid
Both pairs of opp. sides are ll
Exactly 1 pair of opp. Sides are ll
Diagonals are perp.
Diagonals are cong.
Diagonals bisect each other X X X X X X X X X X X X

71. Determine whether the statement is true or false
Determine whether the statement is true or false. If it is true, explain why. If it is false, sketch a counterexample.
If CDEF is a kite, then CDEF is a convex polygon.
If GHIJ is a kite, then GHIJ is not a trapezoid.
The number of acute angles in a trapezoid is always either 1 or 2.

72. Assignments
pp # 3-24, 28-34, (odd in class; even for homework)
pp # (odd in class; even for homework)

73. 6.7 Areas of Triangles and Quadrilaterals
Geometry
Spring 2014

74. Area Postulates Area of a Square Postulate Area Congruence Postulate
The area of a square is the square of the length of its sides, or A = s2.
Area Congruence Postulate
If two polygons are congruent, then they have the same area.
Area Addition Postulate
The area of a region is the sum of the areas of its non-overlapping parts.

75. Area Rectangle: A = bh Parallelogram: A = bh Triangle: A = ½ bh
Trapezoid: A = ½ h(b1+b2)
Kite: A = ½ d1d2
Rhombus: A = ½ d1d2

76. Find the area of ∆ ABC. C 7 4 6 L B A 5

77. Find the area of a trapezoid with vertices at A(0,0), B(2,4), C(6,4), and D(9,0).

78. Find the area of the figures.4 L L L L 4 4 2 L L L L 4 5 8 12

79. Find the area of ABCD. ABCD is a parallelogram Area = bh = (16)(9)
= 144 9 E 16 A D 12

80. Find the area of a trapezoid.
Find the area of a trapezoid WXYZ with W(8,1), X(1,1), Y(2,5), and Z(5,5).

81. Find the area of rhombus.
Find the area of rhombus ABCD. B
Area of Rhombus
A = ½ d1 d2
= ½ (40)(30)
= ½ (1200)
= 600 15 20 20 A C 15 25 D

82. The area of the kite is160.
Find the length of BD. A 10 D B C

83. Ch 6 Review
Geometry
Spring 2014

84. Review 1 A polygon with 7 sides is called a ____. A) nonagon
B) dodecagon
C) heptagon
D) hexagon
E) decagon

85. Review 2 B A C D Find m<A A) 65° B) 135° C) 100° D) 90° E) 105°
165° C
30°
65° D

86. Review 3 Opposite angles of a parallelogram must be _______.
A) complementary
B) supplementary
C) congruent
D) A and C
E) B and C

87. Review 4
If a quadrilateral has four equal sides, then it must be a _______.
A) rectangle
B) square
C) rhombus
D) A and B
E) B and C

88. Review 5
The perimeter of a square MNOP is 72 inches, and NO = 2x What is the value of x?
A) 15
B) 12
C) 6
D) 9
E) 18

89. Review 6 ABCD is a trapezoid. Find the length of midsegment EF. A) 513 A E 11 B 5 D F C 9

90. Review 7 The quadrilateral below is most specifically a __________.
A) rhombus
B) rectangle
C) kite
D) parallelogram
E) trapezoid

91. Review 8
Find the base length of a triangle with an area of 52 cm2 and a height of 13cm.
A) 8 cm
B) 16 cm
C) 4 cm
D) 2 cm
E) 26 cm

92. Review 9
A right triangle has legs of 24 units and 18 units. The length of the hypotenuse is ____.
A) 15 units
B) 30 units
C) 45 units
D) units
E) 32 units

93. Review 10
Sketch a concave pentagon.
Sketch a convex pentagon.

94. Review 11 What type of quadrilateral is ABCD? Explain your reasoning.
120° A
60° C
120°
Isosceles Trapezoid
Isosceles : AD = BC
Trapezoid : AB ll CD
60° B