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Section 6.1 The Polygon Angle-Sum Theorem

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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 ... – PowerPoint PPT presentation

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Title: Section 6.1 The Polygon Angle-Sum Theorem


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 1980. 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 2880. 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 mltY in pentagon TODAY?

16
Section 6.1 The Polygon Angle-Sum Theorem
  • Problem 3b
  • What is mltG 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 mlt1 in the regular octagon
    below?

19
  • 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.

24
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25
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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

27
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28
  • 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.

30
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31
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32
  • Angles of a polygon that share a side are
    consecutive angles. In the diagram, ltA and ltB
    are consecutive angles because the share side AB.

33

34
  • Problem 1
  • What is ltP in Parallelogram PQRS?

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

36
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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?

39
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40
  • Problem 3

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

42
  • 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 mltC 65.
    What are mltD, mltE, and mltF?

115
Section 6.6 Trapezoids and Kites
  • Problem 1b
  • PQRS is an isosceles trapezoid and mltR 106.
    What are mltP, mltQ, and mltS?

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 trianglesWhat 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 mlt1, mlt2,
    mlt3?

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
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