1. Section 6.1 The Polygon Angle-Sum Theorem
Students will be able to:
Find the sum of the measures of the interior angles of a polygon.
Find the sum of the measures of the exterior angles of a polygon.
Lesson Vocabulary
Equilateral polygon
Equiangular polygon
Regular polygon

2. Section 6.1 The Polygon Angle-Sum Theorem
List the names of all of the polygons with
3 sides to 13 sides:
3 sided: ____________ 8 sided: ____________
4 sided: ____________ 9 sided: ___________
5 sided: ____________ 10 sided: ___________
6 sided: ____________ 11 sided: ___________
7 sided: ____________ 12 sided: ___________
13 sided: ___________

3. Section 6.1 The Polygon Angle-Sum Theorem
A diagonal is a segment that connects two nonconsecutive vertices in a polygon!

4. Section 6.1 The Polygon Angle-Sum Theorem
The Solve It is related to a formula for the sum of the interior angle measures of a CONVEX polygon.

5. Section 6.1 The Polygon Angle-Sum Theorem
Essential Understanding:
The sum of the interior angle measures of a polygon depends on the number of sides the polygon has.
By dividing a polygon with n sides into (n – 2) triangles, you can show that the sum of the interior angle measures of any polygon is a multiple of 180.

6. Section 6.1 The Polygon Angle-Sum Theorem
Problem 1: Finding a Polygon Angle Sum
What is the sum of the interior angle measures of a heptagon?

7. Section 6.1 The Polygon Angle-Sum Theorem
Problem 1b: Finding a Polygon Angle Sum
What is the sum of the interior angle
measures of a 17-gon?

8. Section 6.1 The Polygon Angle-Sum Theorem
Problem 1c:
The sum of the interior angle measures of a polygon is How can you find the number of sides in the polygon? Classify it!

9. Section 6.1 The Polygon Angle-Sum Theorem
Problem 1d:
The sum of the interior angle measures of a polygon is How can you find the number of sides in the polygon? Classify it!!!

10. Section 6.1 The Polygon Angle-Sum Theorem

11. Section 6.1 The Polygon Angle-Sum Theorem

12. Section 6.1 The Polygon Angle-Sum Theorem
Problem 2:
What is the measure of each interior angle in a regular hexagon?

13. Section 6.1 The Polygon Angle-Sum Theorem
Problem 2b:
What is the measure of each interior angle in a regular nonagon?

14. Section 6.1 The Polygon Angle-Sum Theorem
Problem 2c:
What is the measure of each interior angle in a regular 100-gon?
Explain what happens to the interior angles of a regular figure the more sides the figure has? What is the value approaching but will never get to?

15. Section 6.1 The Polygon Angle-Sum Theorem
Problem 3:
What is m<Y in pentagon TODAY?

16. Section 6.1 The Polygon Angle-Sum Theorem
Problem 3b:
What is m<G in quadrilateral EFGH?

17. Section 6.1 The Polygon Angle-Sum Theorem
You can draw exterior angles at any vertex of a polygon. The figures below show that the sum of the measures of exterior angles, one at each vertex, is 360.

18. Problem 4: What is m<1 in the regular octagon below?

19. What is the measure of an exterior angle of a regular pentagon?
Problem 4b:
What is the measure of an exterior angle of a regular pentagon?

20. Problem 5:
What do you notice about the sum of the interior angle and exterior angle of a regular figure?

21. Problem 6:
If the measure of an exterior angle of a regular polygon is 18. Find the measure of the interior angle. Then find the number of sides the polygon has.

22. Problem 6b:
If the measure of an exterior angle of a regular polygon is 72. Find the measure of the interior angle. Then find the number of sides the polygon has.

23. Problem 6c:
If the measure of an exterior angle of a regular polygon is x. Find the measure of the interior angle. Then find the number of sides the polygon has.

26. Section 6.2 – Properties of Parallelograms
Students will be able to:
Use relationships among sides and angles of parallelograms
Use relationships among diagonals of parallograms
Lesson Vocabulary:
Parallelogram
Opposite Angles
Opposite Sides
Consecutive Angles

28. Essential Understanding:
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
Essential Understanding:
Parallelograms have special properties regarding their sides, angles, and diagonals.

29. In a quadrilateral, opposite sides do not share a vertex and opposite angles do not share a side.

32. Angles of a polygon that share a side are consecutive angles
Angles of a polygon that share a side are consecutive angles. In the diagram, <A and <B are consecutive angles because the share side AB.

34. Problem 1: What is <P in Parallelogram PQRS?

35. Problem 1b: Find the value of x in each parallelogram.

37. Problem 2:
Solve a system of linear equations to find the values of x and y in Parallelogram KLMN. What are KM and LN?

38. Problem 2b:
Solve a system of linear equations to find the values of x and y in Parallelogram PQRS. What are PR and SQ?

40. Problem 3:

41. Find the value(s) of the variable(s) in each parallelogram.
Extra Problems:
Find the value(s) of the variable(s) in each parallelogram.

42. Find the measures of the numbered angles for each parallelogram.
Extra Problems:
Find the measures of the numbered angles for each parallelogram.

43. Extra Problems:

44. Extra Problems:

45. Extra Problems:

46. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram
Students will be able to:
Determine whether a quadrilateral is a parallelogram

47. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram
Essential Understanding:
You can decide whether a quadrilateral is a parallelogram if its sides, angles, and diagonals have certain properties.
In Lesson 6-2, you learned theorems about the properties of parallelograms. In this lesson, you will learn the converses of those theorems. That is, if a quadrilateral has certain properties, then it must be a parallelogram.

48. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

49. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

50. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

51. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

52. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

53. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram
Problem 1:
For what value of y must PQRS be a parallelogram?

54. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram
Problem 1b:
For what value of x and y must ABCD be a parallelogram?

55. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram
Problem 1c:
For what value of x and y must ABCD be a parallelogram?

56. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram
Problem 1d:
For what value of x and y must ABCD be a parallelogram?

57. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram
Problem 1e:
For what value of x and y must ABCD be a parallelogram?

58. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram
Problem 2:
Can you prove that the quadrilateral is a parallelogram based on the given information? Explain!

59. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram
Problem 2:
Can you prove that the quadrilateral is a parallelogram based on the given information? Explain!

60. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram
Problem 3:
A truck sits on the platform of a vehicle lift. Two moving arms raise the platform until a mechanic can fit underneath. Why will the truck always remain parallel to the ground as it is lifted? Explain!

61. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

62. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

63. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

64. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

65. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Students will be able to:
Define and Classify special types of parallelograms
Use the Properties of Rhombuses and Rectangles
Lesson Vocabulary
Rhombus
Rectangle
Square

66. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

67. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
A rhombus is a parallelogram with
four congruent sides.

68. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
A rectangle is a parallelogram with
four right angles.

69. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
A square is a parallelogram with
four congruent sides and four right angles.

70. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

71. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Problem 1:
Is Parallelogram ABCD a rhombus, rectangle or square? Explain!

72. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Problem 1b:
Is Parallelogram EFGH a rhombus, rectangle or square? Explain!

73. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

74. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

75. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Problem 2:
What are the measures of the numbered angles in rhombus ABCD?

76. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Problem 2:
What are the measures of the numbered angles in rhombus PQRS?

77. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Problem 2:
What are the measures of the numbered angles in the rhombus?

78. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Problem 2:
What are the measures of the numbered angles in the rhombus?

79. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

80. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Problem 3:
In rectangle RSBF, SF = 2x + 15 and
RB = 5x – 12. What is the length of a diagonal?

81. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Problem 4:
LMNP is a rectangle. Find the value of x and the length of each diagonal
LN = 5x – 8 and MP = 2x + 1

82. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Problem 5:
Determine the most precise name for each quadrilateral.

83. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Problem 6:
List all quadrilaterals that have the given property. Chose among parallelogram, rhombus, rectangle, or square.
Opposite angles are congruent.

84. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Problem 6b:
List all quadrilaterals that have the given property. Chose among parallelogram, rhombus, rectangle, or square.
Diagonals are congruent

85. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Problem 6c:
List all quadrilaterals that have the given property. Chose among parallelogram, rhombus, rectangle, or square.
Each diagonal bisects opposite angles

86. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares
Problem 6d:
List all quadrilaterals that have the given property. Chose among parallelogram, rhombus, rectangle, or square.
Opposite sides are parallel

87. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

88. Section 6.4 – Properties of Rhombuses, Rectangles, and Squares

89. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Students will be able to:
Determine whether a parallelogram is a rhombus or rectangle.

90. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

91. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

92. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

93. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

94. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Problem 1:
Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain!

95. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Problem 1b:
Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain!

96. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Problem 1c:
Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain!

97. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Problem 1d:
Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain!

98. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Problem 2:
For what value of x is parallelogram ABCD a rhombus?

99. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Problem 2b:
For what value of x is the parallelogram a rectangle?

100. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Problem 2c:
For what value of x is the parallelogram a rhombus?

101. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Problem 2d:
For what value of x is the parallelogram a rectangle?

102. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Problem 2e:
For what value of x is the parallelogram a rectangle?

103. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Problem 3:
Builders use properties of diagonals to “square up” rectangular shapes like building frames and playing-field boundaries. Suppose you are on the volunteer building team at the right. You are helping to lay out a rectangular patio for a youth center. How can you use the properties of diagonals to locate the four corners?

104. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Problem 4:
Determine whether the quadrilateral can be a parallelogram. Explain!
The diagonals are congruent, but the quadrilateral has no right angles.

105. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Problem 4b:
Determine whether the quadrilateral can be a parallelogram. Explain!
Each diagonal is 3 cm long and two opposite sides are 2 cm long.

106. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares
Problem 4c:
Determine whether the quadrilateral can be a parallelogram. Explain!
Two opposite angles are right angles but the quadrilateral is not a rectangle.

107. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

108. Section 6.5 – Conditions for Rhombuses, Rectangles, and Squares

109. Section 6.6 – Trapezoids and Kites
Students will be able to:
Verify and use properties of trapezoids and kites.
Lesson Vocabulary
Trapezoid
Base
Leg
Base angle
Isosceles trapezoid
Midsegment of a trapezoid
Kite

110. Section 6.6 – Trapezoids and Kites
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
The parallel sides are of trapezoid are called bases.
The nonparallel sides are called legs.
The two angles that share a base of a trapezoid are called base angles. A trapezoid has two pairs of base angles.

111. Section 6.6 – Trapezoids and Kites
An isosceles trapezoid is a trapezoid with legs that are congruent. ABCD below is an isosceles trapezoid. The angles of an isosceles trapezoid have some unique properties.

112. Section 6.6 – Trapezoids and Kites

113. Section 6.6 – Trapezoids and Kites

114. Section 6.6 – Trapezoids and Kites
Problem 1:
CDEF is an isosceles trapezoid and m<C = 65. What are m<D, m<E, and m<F?

115. Section 6.6 – Trapezoids and Kites
Problem 1b:
PQRS is an isosceles trapezoid and m<R = What are m<P, m<Q, and m<S?

116. Section 6.6 – Trapezoids and Kites
Problem 2:
The second ring of the paper fan consists of 20 congruent isosceles trapezoids that appear to form circles. What are the measures of the base angles of these trapezoids?

117. Section 6.6 – Trapezoids and Kites
Problem 3:
Find the measures of the numbered angles in each isosceles trapezoid.

118. Section 6.6 – Trapezoids and Kites
Problem 3b:
Find the measures of the numbered angles in each isosceles trapezoid.

119. Section 6.6 – Trapezoids and Kites
Problem 3c:
Find the measures of the numbered angles in each isosceles trapezoid.

120. Section 6.6 – Trapezoids and Kites
In lesson 5.1 you learned about the midsegments of triangles…What are they????
Trapezoids also have midsegments.
The midsegment of a trapezoid is the segment that joins the midpoints of its legs. The midsegment has two unique properties.

121. Section 6.6 – Trapezoids and Kites

122. Section 6.6 – Trapezoids and Kites
Problem 4:
Segment QR is the midsegment of trapezoid LMNP. What is x?

123. Section 6.6 – Trapezoids and Kites
Problem 4b:
Find EF is the trapezoid.

124. Section 6.6 – Trapezoids and Kites
Problem 4c:
Find EF is the trapezoid.

125. Section 6.6 – Trapezoids and Kites
Problem 4e:
Find the lengths of the segments with variable expressions.

126. Section 6.6 – Trapezoids and Kites
A kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent.

127. Section 6.6 – Trapezoids and Kites

128. Section 6.6 – Trapezoids and Kites
Problem 5:
Quadrilateral DEFG is a kite. What are m<1, m<2, m<3?

129. Section 6.6 – Trapezoids and Kites
Problem 5b:
Find the measures of the numbered angles in each kite.

130. Section 6.6 – Trapezoids and Kites
Problem 5c:
Find the measures of the numbered angles in each kite.

131. Section 6.6 – Trapezoids and Kites
Problem 5d:
Find the measures of the numbered angles in each kite.

132. Section 6.6 – Trapezoids and Kites
Problem 5e:
Find the value(s) of the variable(s) in each kite.

133. Section 6.6 – Trapezoids and Kites
Problem 5f:
Find the value(s) of the variable(s) in each kite.

134. Section 6.6 – Trapezoids and Kites

135. Section 6.6 – Trapezoids and Kites
Problem 6:
Determine whether each statement is true or false. Be able to justify your answer.
All squares are rectangles
A trapezoid is a parallelogram
A rhombus can be a kite
Some parallelograms are squares
Every quadrilateral is a parallelogram
All rhombuses are squares.

136. Section 6.6 – Trapezoids and Kites
Problem 7:
Name each type of quadrilateral that can meet the given condition.
Exactly one pair of congruent sides
Two pairs of parallel sides
Four right angles
Adjacent sides that are congruent
Perpendicular diagonals
Congruent diagonals

137. Section 6.6 – Trapezoids and Kites
Problem 8:
Can two angles of a kite be as follows? Explain!
Opposite and acute
Consecutive and obtuse
Opposite and supplementary
Consecutive and supplementary
Opposite and complementary
Consecutive and complementary

138. Section 6.6 – Trapezoids and Kites
Problem 8:
Can two angles of a kite be as follows? Explain!
Opposite and acute
Consecutive and obtuse
Opposite and supplementary
Consecutive and supplementary
Opposite and complementary
Consecutive and complementary

139. Section 6.6 – Trapezoids and Kites

140. Section 6.6 – Trapezoids and Kites