Title: Section 6.1 The Polygon Angle-Sum Theorem
1Section 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
2Section 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 ___________
3Section 6.1 The Polygon Angle-Sum Theorem
- A diagonal is a segment that connects two
nonconsecutive vertices in a polygon!
4Section 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.
5Section 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.
6Section 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?
7Section 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?
8Section 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!
9Section 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!!!
10Section 6.1 The Polygon Angle-Sum Theorem
11Section 6.1 The Polygon Angle-Sum Theorem
12Section 6.1 The Polygon Angle-Sum Theorem
- Problem 2
- What is the measure of each interior angle in a
regular hexagon?
13Section 6.1 The Polygon Angle-Sum Theorem
- Problem 2b
- What is the measure of each interior angle in a
regular nonagon?
14Section 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?
15Section 6.1 The Polygon Angle-Sum Theorem
- Problem 3
- What is mltY in pentagon TODAY?
16Section 6.1 The Polygon Angle-Sum Theorem
- Problem 3b
- What is mltG in quadrilateral EFGH?
17Section 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(No Transcript)
25(No Transcript)
26Section 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(No Transcript)
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(No Transcript)
31(No Transcript)
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(No Transcript)
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(No Transcript)
40 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 44 45 46Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
- Students will be able to
- Determine whether a quadrilateral is a
parallelogram
47Section 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.
48Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
49Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
50Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
51Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
52Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
53Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
- Problem 1
- For what value of y must PQRS be a parallelogram?
54Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
- Problem 1b
- For what value of x and y must ABCD be a
parallelogram?
55Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
- Problem 1c
- For what value of x and y must ABCD be a
parallelogram?
56Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
- Problem 1d
- For what value of x and y must ABCD be a
parallelogram?
57Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
- Problem 1e
- For what value of x and y must ABCD be a
parallelogram?
58Section 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!
59Section 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!
60Section 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!
61Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
62Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
63Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
64Section 6.3 Proving That a Quadrilateral Is a
Parallelogram
65Section 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
66Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
67Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
- A rhombus is a parallelogram with
- four congruent sides.
68Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
- A rectangle is a parallelogram with
- four right angles.
69Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
- A square is a parallelogram with
- four congruent sides and four right angles.
70Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
71Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
- Problem 1
- Is Parallelogram ABCD a rhombus, rectangle or
square? Explain!
72Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
- Problem 1b
- Is Parallelogram EFGH a rhombus, rectangle or
square? Explain!
73Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
74Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
75Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
- Problem 2
- What are the measures of the numbered angles in
rhombus ABCD?
76Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
- Problem 2
- What are the measures of the numbered angles in
rhombus PQRS?
77Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
- Problem 2
- What are the measures of the numbered angles in
the rhombus?
78Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
- Problem 2
- What are the measures of the numbered angles in
the rhombus?
79Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
80Section 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?
81Section 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
82Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
- Problem 5
- Determine the most precise name for each
quadrilateral.
83Section 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.
84Section 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
85Section 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
86Section 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
87Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
88Section 6.4 Properties of Rhombuses,
Rectangles, and Squares
89Section 6.5 Conditions for Rhombuses,
Rectangles, and Squares
- Students will be able to
- Determine whether a parallelogram is a rhombus or
rectangle.
90Section 6.5 Conditions for Rhombuses,
Rectangles, and Squares
91Section 6.5 Conditions for Rhombuses,
Rectangles, and Squares
92Section 6.5 Conditions for Rhombuses,
Rectangles, and Squares
93Section 6.5 Conditions for Rhombuses,
Rectangles, and Squares
94Section 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!
95Section 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!
96Section 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!
97Section 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!
98Section 6.5 Conditions for Rhombuses,
Rectangles, and Squares
- Problem 2
- For what value of x is parallelogram ABCD a
rhombus?
99Section 6.5 Conditions for Rhombuses,
Rectangles, and Squares
- Problem 2b
- For what value of x is the parallelogram a
rectangle?
100Section 6.5 Conditions for Rhombuses,
Rectangles, and Squares
- Problem 2c
- For what value of x is the parallelogram a
rhombus?
101Section 6.5 Conditions for Rhombuses,
Rectangles, and Squares
- Problem 2d
- For what value of x is the parallelogram a
rectangle?
102Section 6.5 Conditions for Rhombuses,
Rectangles, and Squares
- Problem 2e
- For what value of x is the parallelogram a
rectangle?
103Section 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?
104Section 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.
105Section 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.
106Section 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.
107Section 6.5 Conditions for Rhombuses,
Rectangles, and Squares
108Section 6.5 Conditions for Rhombuses,
Rectangles, and Squares
109Section 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
110Section 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.
111Section 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.
112Section 6.6 Trapezoids and Kites
113Section 6.6 Trapezoids and Kites
114Section 6.6 Trapezoids and Kites
- Problem 1
- CDEF is an isosceles trapezoid and mltC 65.
What are mltD, mltE, and mltF?
115Section 6.6 Trapezoids and Kites
- Problem 1b
- PQRS is an isosceles trapezoid and mltR 106.
What are mltP, mltQ, and mltS?
116Section 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?
117Section 6.6 Trapezoids and Kites
- Problem 3
- Find the measures of the numbered angles in each
isosceles trapezoid.
118Section 6.6 Trapezoids and Kites
- Problem 3b
- Find the measures of the numbered angles in each
isosceles trapezoid.
119Section 6.6 Trapezoids and Kites
- Problem 3c
- Find the measures of the numbered angles in each
isosceles trapezoid.
120Section 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.
121Section 6.6 Trapezoids and Kites
122Section 6.6 Trapezoids and Kites
- Problem 4
- Segment QR is the midsegment of trapezoid LMNP.
What is x?
123Section 6.6 Trapezoids and Kites
- Problem 4b
- Find EF is the trapezoid.
124Section 6.6 Trapezoids and Kites
- Problem 4c
- Find EF is the trapezoid.
125Section 6.6 Trapezoids and Kites
- Problem 4e
- Find the lengths of the segments with variable
expressions.
126Section 6.6 Trapezoids and Kites
- A kite is a quadrilateral with two pairs of
consecutive sides congruent and no opposite sides
congruent.
127Section 6.6 Trapezoids and Kites
128Section 6.6 Trapezoids and Kites
- Problem 5
- Quadrilateral DEFG is a kite. What are mlt1, mlt2,
mlt3?
129Section 6.6 Trapezoids and Kites
- Problem 5b
- Find the measures of the numbered angles in each
kite.
130Section 6.6 Trapezoids and Kites
- Problem 5c
- Find the measures of the numbered angles in each
kite.
131Section 6.6 Trapezoids and Kites
- Problem 5d
- Find the measures of the numbered angles in each
kite.
132Section 6.6 Trapezoids and Kites
- Problem 5e
- Find the value(s) of the variable(s) in each
kite.
133Section 6.6 Trapezoids and Kites
- Problem 5f
- Find the value(s) of the variable(s) in each
kite.
134Section 6.6 Trapezoids and Kites
135Section 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.
136Section 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
137Section 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
138Section 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
139Section 6.6 Trapezoids and Kites
140Section 6.6 Trapezoids and Kites