Transcript polygon - Mona Shores Blogs

Chapter 6
Quadrilaterals
Chapter Objectives

Define a polygon and its characteristics
 Identify a regular polygon
 Interior Angles of a Quadrilateral Theorem
 Properties of Parallelograms
 Using coordinate geometry to prove parallelograms
 Compare rhombuses, rectangles, and squares
 Identify trapezoids and kites
 Midsegment Theorem for Trapezoids
 Calculate area of trapezoids, kites, rhombuses,
rectangles, and squares
Lesson 6.1
Polygons
Lesson 6.1 Objectives
 Identify
a figure to be a polygon.
 Recognize the different types of
polygons based on the number of sides.
 Identify the components of a polygon.
 Use the sum of the interior angles of a
quadrilateral.
Definition of a Polygon

A polygon is plane figure (two-dimensional)
that meets the following conditions.
1. It is formed by three or more segments called sides.
2. The sides must be straight lines.
3. Each side intersects exactly two other sides, one at each
endpoint.
4. The polygon is closed in all the way around with no gaps.
5. Each side must end when the next side begins. No tails.
Polygons
Not Polygons
Polygon Parts

Each segment that is used to close a polygon
in is called a side.
 Where each side ends is called a vertex.

A vertex is simply a corner of the polygon.
vertices
sides
Types of Polygons
Number of Sides
3
4
5
6
7
8
9
10
12
n
Type of Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
n-gon
Concave v Convex


A polygon is convex if no
line that contains a side of
the polygon contains a point
in the interior of the polygon.
Take any two points in the
interior of the polygon. If you
can draw a line between the
two points that never leave
the interior of the polygon,
then it is convex.



A polygon is concave if a
line that contains a side of
the polygon contains a point
in the interior of the polygon.
Take any two points in the
interior of the polygon. If you
can draw a line between the
two points that does leave
the interior of the polygon,
then it is concave.
Concave polygons have
dents in the sides, or you
could say it caves in.
Example 1
Determine if the following are polygons or not.
If it is a polygon, classify it as concave or convex.
No!
Yes
Yes
Concave
Convex
Regular Polygons



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 both equilateral and equiangular.
The best way to draw these is to label each sides and angle with the
proper congruent marks.
Diagonals of a Polygon

A diagonal of a polygon is a segment that
joins two nonconsecutive vertices.

A diagonal does not go to the point next to it.


That would make it a side!
Diagonals cut across the polygon to all points on
the other side.

There is typically more than one diagonal.
Theorem 6.1:
Interior Angles of a Quadrilateral Theorem

The sum of the measures of the interior angles of a
quadrilateral is 360o.
4
3
1
2
m 1 +m 2 + m 3 + m 4 = 360o
Homework 6.1
 In
Class
 1-11

p325-328
 HW
 12-46,
 Due
54-59
Tomorrow
Lesson 6.2
Properties of Parallelograms
Lesson 6.2 Objectives
 Define
a parallelogram
 Identify properties of parallelograms
 Use properties of parallelograms to
determine unknown quantities of the
parallelogram
Definition of a Parallelogram

A parallelogram is a quadrilateral with both pairs of
opposite sides parallel.
Theorem 6.2:
Congruent Sides of a Parallelogram

If a quadrilateral is a parallelogram, then its
opposite sides are congruent.
Theorem 6.3:
Opposite Angles of a Parallelogram

If a quadrilateral is a parallelogram, then its
opposite angles are congruent.
Example 2
Find the missing variables in the parallelograms.


x = 11
y=8
d = 53
d + 15 = 68
 
c – 5 = 20
c = 25

m = 101
Theorem 6.4:
Consecutive Angles of a Parallelogram

If a quadrilateral is a parallelogram, then its
consecutive angles are supplementary.
Q
R
P
S
m P + m S = 180o
m Q + m R = 180o
m P + m Q = 180o
m R + m S = 180o
Theorem 6.5:
Diagonals of a Parallelogram

If a quadrilateral is a parallelogram, then its
diagonals bisect each other.

Remember that means to cut into two congruent
segments.
Example 3
Find the indicated measure in  HIJK
a)
HI
16
a)
a)
Theorem 6.2
GH
b)
8
b)
b)
Theorem 6.6
KH
c)
10
c)
c)
Theorem 6.2
HJ
d)
16
d)
d)
Theorem 6.6 & Seg Add Post
m KIH
e)
28o
e)
e)
AIA Theorem
m JIH
f)
96o
f)
Theorem 6.4
f)
m KJI
g)
g)
84o
g)
Theorem 6.3
Homework 6.2
 HW

p333-336
 20-37,
47-54, 60, 61
 Due
Tomorrow
 Quiz Wednesday
 Lessons
6.1-6.3
Lesson 6.3
Proving Quadrilaterals
are
Parallelograms
Lesson 6.3 Objectives
 Verify
that a quadrilateral is a
parallelogram.
 Utilize coordinate geometry with
parallelograms
Theorem 6.6:
Congruent Sides of a Parallelogram Converse

If both pairs of opposite sides are congruent,
then it is a parallelogram.
Theorem 6.7:
Opposite Angles of a Parallelogram Converse

If both pairs of opposite angles are
congruent, then it is a parallelogram.
Theorem 6.8:
Consecutive Angles of a Parallelogram Converse

If an angle of a quadrilateral is
supplementary to its consecutive angles,
then it is a parallelogram.
Q
R
P
S
m P + m S = 180o
m Q + m R = 180o
m P + m Q = 180o
m R + m S = 180o
Theorem 6.9:
Diagonals of a Parallelogram Converse

If the diagonals of a quadrilateral bisect each
other, then it is a parallelogram.
Theorem 6.10:
Opposite Sides of a Parallelogram
 If
one pair of opposite sides of a
quadrilateral are congruent and
parallel, then the quadrilateral is a
parallelogram.
Example 4
Which theorem would you use to show the following are parallelograms?
Theorem 6.10
Theorem 6.9
Theorem 6.6
Theorem 6.6
or
Theorem 6.10
Theorem 6.7
Theorem 6.8
or
Theorem 6.7
Homework 6.3

In Class

1-7


p342-345
HW

9-29, 45-47

skip 15-16

Due Tomorrow
 Quiz Friday

Lessons 6.1-6.3
Lesson 6.4
Rhombuses,
Rectangles,
and
Squares
Lesson 6.4 Objectives
 Identify
characteristics of a rhombus.
 Identify characteristics of a rectangle.
 Identify characteristics of a square.
Rhombus

A rhombus is a parallelogram with four congruent
sides.

The rhombus corollary states that a quadrilateral is a
rhombus if and only if it has four congruent sides.
Theorem 6.11:
Perpendicular Diagonals
 A parallelogram
is a rhombus if and only
if its diagonals are perpendicular.
Theorem 6.12:
Opposite Angle Bisector

A parallelogram is a rhombus iff each
diagonal bisects a pair of opposite angles.
Rectangle

A rectangle is a parallelogram with four
congruent angles.

The rectangle corollary states that a quadrilateral
is a rectangle iff it has four right angles.
Theorem 6.13:
Four Congruent Diagonals
 A parallelogram
is a rectangle iff all four
segments of the diagonals are
congruent.
Square

A square is a parallelogram with four
congruent sides and four congruent angles.
Square Corollary
 A quadrilateral
is a square iff it s a
rhombus and a rectangle.
 So that means that all the properties of
rhombuses and rectangles work for a
square at the same time.
Example 5
Classify the parallelogram.
Explain your reasoning.
Must be
supplementary



Rhombus
Square
Rectangle
Diagonals are perpendicular.
Theorem 6.11
Square Corollary
Diagonals are congruent.
Theorem 6.13
Homework 6.4
 In
Class
 1,

3-11
p351-354
 HW
 12-46
 Due
evens, 55-58, 66, 67
Tomorrow
Lesson 6.5
Trapezoids
and
Kites
Lesson 6.5 Objectives
 Identify
properties of a trapezoid.
 Recognize an isosceles trapezoid.
 Utilize the midsegment of a trapezoid to
calculate other quantities from the
trapezoid.
 Identify a kite.
Trapezoid

A trapezoid is a quadrilateral with exactly one pair of
parallel sides.
 The parallel sides are called the bases.
 The nonparallel sides are called legs.
 The angles formed by the bases are called the
base angles.
Isosceles Trapezoid
 If
the legs of a trapezoid are congruent,
then the trapezoid is an isosceles
trapezoid.
Theorem 6.14:
Bases Angles of a Trapezoid

If a trapezoid is isosceles, then each pair of
base angles is congruent.


That means the top base angles are congruent.
The bottom base angles are congruent.

But they are not all congruent to each other!
Theorem 6.15:
Base Angles of a Trapezoid Converse

If a trapezoid has one pair of congruent base
angles, then it is an isosceles trapezoid.
Theorem 6.16:
Congruent Diagonals of a Trapezoid

A trapezoid is isosceles if and only if its
diagonals are congruent.

Notice this is the entire diagonal itself.

Don’t worry about it being bisected cause it’s not!!
Example 6
Find the measures of the other three angles.
127o


53o
127o
Supplementary
because of CIA
83o


97o
83o
Supplementary
because of CIA
Midsegment
 The
midsegment of a trapezoid is the
segment that connects the midpoints of
the legs of a trapezoid.
Theorem 6.17:
Midsegment Theorem for Trapezoids

The midsegment of a trapezoid is parallel to
each base and its length is one half the sum
of the lengths of the bases.
C

It is the average of the base lengths!
D
N
M
MN = 1/2(AB + CD)
A
B
Example 7
Find the length of the midsegment.
RT = 1/2(WX + ZY)
RT = 1/2(7 + 13)
RT = 1/2(20)
RT = 10
RT = 1/2(WX + ZY)
RT = 1/2(9 + 12)
RT = 1/2(21)
RT = 10.5
Kite

A kite is a quadrilateral that has two pairs of
consecutive sides that are congruent, but
opposite sides are not congruent.


It looks like the kite you got for your birthday when
you were 5!
There are no sides that are parallel.
Theorem 6.18:
Diagonals of a Kite

If a quadrilateral is a kite, then its diagonals
are perpendicular.
Theorem 6.19:
Opposite Angles of a Kite

If a quadrilateral is a kite, then exactly one
pair of opposite angles are congruent.

The angles that are congruent are between the
two different congruent sides.

You could call those the shoulder angles.
NOT
Example 8
Find the missing angle measures.
64o

125o
88o
125o
60 + K + 50 + M = 360
88 + 120 + 88 + J = 360
60 + M + 50 + M = 360
296 + J = 360
110 + 2M = 360
J = 64
2M = 250
M = 125
K = 125
But K  M
Use Pythagorean Theorem!
Example 9
Cause the diagonals are perpendicular!!
a2 + b2 = c2
Find the lengths of all the sides of the kite.
Round your answer to the nearest hundredth.
a2 + b2 = c2
52 + 52 = c2
25 + 25 = c2
7.07
7.07
50 = c2
a2 + b2 = c2
52 + 122 = c2
25 + 144 = c2
c = 7.07
13
13
169 = c2
c = 13
Homework 6.5
 In
Class
 3-9

p359-362
 HW
 10-39,
51, 52, 57-64
 Due
Tomorrow
 Test Monday
 November
12
Lesson 6.6
Special Quadrilaterals
Lesson 6.6 Objectives
 Create
a hierarchy of polygons
 Identify special quadrilaterals based on
limited information
Polygon Hierarchy
Polygons
Triangles
Parallelogram
Rhombus
Quadrilaterals
Trapezoid
Rectangle
Square
Pentagons
Kite
Isosceles Trapezoid
NEVER
How to Read the Hierarchy
Polygons
Parallelogram
Rhombus
Quadrilaterals
Trapezoid
Rectangle
Square
So that means that a square is always a
rhombus, a parallelogram, a quadrilateral,
and a polygon.
Pentagons
Kite
Isosceles Trapezoid
But a parallelogram is sometimes a
rhombus and sometimes a square.
However, a parallelogram is never a
trapezoid or a kite.
SOMETIMES
ALWAYS
Triangles
Using the Hierarchy

Remember that a square must fit all the
properties of its “ancestors.”


That means the properties of a rhombus,
rectangle, parallelogram, quadrilateral, and
polygon must all be true!
So when asked to identify a figure as specific
as possible, test the properties working your
way down the hierarchy.

As soon as you find a figure that doesn’t work any
more you should be able to identify the specific
name of that figure.
Homework 6.6
 In
Class
 2-7

p367-370
 HW
 8-35,
55-65
 Due
Tomorrow
 Test Friday
 November
7
Lesson 6.7
Areas of
Triangles
and
Quadrilaterals
Lesson 6.7 Objectives
 Find
the area of any type of triangle.
 Find the area of any type of
quadrilateral.
Postulate 22:
Area of a Square Postulate
 The
area of a square is the square of
the length of its side.
A
= s2
s
Area Postulates

Postulate 23: Area
Congruence
Postulate

If two polygons are
congruent, then they
have the same area.

Postulate 24: Area
Addition Postulate

The area of a region
is the sum of the
areas of its
nonoverlapping
parts.
Theorem 6.20:
Area of a Rectangle
 The
area of a rectangle is the product of
a base and its corresponding height.
 Corresponding
height indicates a segment
perpendicular to the base to the opposite
side.
A =
bh
h
b
Theorem 6.21:
Area of a Parallelogram

The area of a parallelogram is the product of
a base and its corresponding height.


Remember the height must be perpendicular to
one of the bases.
The height will be given to you or you will need to
find it.

To find it, use Pythagorean Theorem


a2 + b2 = c2
A = bh
h
b
Theorem 6.22:
Area of a Triangle

The area of a triangle is one half the product
of the base and its corresponding height.

The base for this formula is the segment that is
perpendicular to the height.

It may be a side of the triangle, it may not!
h
h
h
b
b
b
Theorem 6.23:
Area of a Trapezoid

The area of a trapezoid is one half the
product of the height and the sum of the
bases.


The height is the perpendicular segment between
the bases of the trapezoid.
A = ½ h (b1+b2)
b1
h
b2
Theorem 6.24:
Area of a Kite
 The
area of a kite is one half the
product of the lengths of the diagonals.
A
= ½ d1d2
d1
d2
Theorem 6.25:
Area of a Rhombus
 The
area of a rhombus is equal to one
half the product of the lengths of the
diagonals.
d
A
= ½ d1d2
1
d2
Homework 6.7
 In
Class
 3-13

p376-380
 HW
 14-38
evens, 50-52, 60, 61
 Due
Tomorrow
 Test Monday
 November
12