Chapter 6 Quadrilaterals
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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.
sidesvertices
Types of PolygonsNumber of Sides Type of Polygon
3
4
5
6
7
8
9
10
12
n
TriangleQuadrilateral
PentagonHexagonHeptagonOctagonNonagonDecagon
Dodecagonn-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 1Determine if the following are polygons or not.
If it is a polygon, classify it as concave or convex.
No! Yes
Concave
Yes
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.
1 2
3 4
m 1 +m 2 + m 3 + m 4 = 360o
Homework 6.1 In Class
1-11 p325-328
HW 12-46, 54-59
Due Tomorrow
Lesson 6.2
Properties of Parallelograms
Lesson 6.2 ObjectivesDefine a parallelogram Identify properties of parallelogramsUse 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 2Find the missing variables in the parallelograms.
x = 11
y = 8
m = 101
c – 5 = 20
c = 25
d + 15 = 68
d = 53
Theorem 6.4:Consecutive Angles of a Parallelogram
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
Q
P
R
S
m P + m S = 180o
m P + m Q = 180o
m Q + m R = 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 3Find the indicated measure in HIJKa) HI
a) 16a) Theorem 6.2
b) GHb) 8
b) Theorem 6.6
c) KHc) 10
c) Theorem 6.2
d) HJd) 16
d) Theorem 6.6 & Seg Add Post
e) m KIHe) 28o
e) AIA Theorem
m JIHa) 96o
a) Theorem 6.4
a) m KJIa) 84o
b) Theorem 6.3
Homework 6.2HW
p333-336 20-37, 47-54, 60, 61
Due TomorrowQuiz Wednesday
Lessons 6.1-6.3
Lesson 6.3
Proving Quadrilaterals
are
Parallelograms
Lesson 6.3 ObjectivesVerify 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
P
R
Sm P + m S = 180o
m P + m Q = 180o
m Q + m R = 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 4Which theorem would you use to show the following are parallelograms?
Theorem 6.10
Theorem 6.9
Theorem 6.6
Theorem 6.6or
Theorem 6.10
Theorem 6.7
Theorem 6.8or
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 CorollaryA 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 5Classify the parallelogram.Explain your reasoning.
RhombusDiagonals are perpendicular.
Theorem 6.11
SquareSquare Corollary
Must be supplementary
RectangleDiagonals are congruent.
Theorem 6.13
Homework 6.4 In Class
1, 3-11 p351-354
HW 12-46 evens, 55-58, 66, 67
Due 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 6Find the measures of the other three angles.
53o Supplementary
because of CIA127o
127o
Supplementarybecause of CIA
97o
83o
83o
MidsegmentThe 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. It is the average of the base lengths!
A B
C D
M N
MN = 1/2(AB + CD)
Example 7Find 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.
88o
88 + 120 + 88 + J = 360
296 + J = 360
J = 64
60 + K + 50 + M = 360But K M
60 + M + 50 + M = 360
110 + 2M = 360
2M = 250
M = 125
K = 125
64o 125o
125o
Example 9Find the lengths of all the sides of the kite.Round your answer to the nearest hundredth.
Use Pythagorean Theorem!
Cause the diagonals are perpendicular!!
a2 + b2 = c2
a2 + b2 = c2
52 + 52 = c2
25 + 25 = c2
50 = c2
c = 7.07
7.07 7.07a2 + b2 = c2
52 + 122 = c2
25 + 144 = c2
169 = c2
c = 131313
Homework 6.5 In Class
3-9 p359-362
HW 10-39, 51, 52, 57-64
Due TomorrowTest Monday
November 12
Lesson 6.6
Special Quadrilaterals
Lesson 6.6 ObjectivesCreate a hierarchy of polygons Identify special quadrilaterals based on
limited information
Polygon HierarchyPolygons
Triangles Quadrilaterals Pentagons
Rhombus Rectangle
TrapezoidParallelogram Kite
Square
Isosceles Trapezoid
How to Read the HierarchyPolygons
Triangles Quadrilaterals Pentagons
Rhombus Rectangle
TrapezoidParallelogram Kite
Square
Isosceles TrapezoidALW
AYS
SOM
ETIM
ES
So that means that a square is always a rhombus, a parallelogram, a quadrilateral, and a polygon.
But a parallelogram is sometimes a rhombus and sometimes a square.
However, a parallelogram is never a trapezoid or a kite.
NEVER
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 TomorrowTest Friday
November 7
Lesson 6.7
Areas of
Triangles
and
Quadrilaterals
Lesson 6.7 ObjectivesFind 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 RectangleThe 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
b
h
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
b
h
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!
b b b
h h h
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)
b2
h
b1
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
d2
d1
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. A = ½ d1d2
d2
d1
Homework 6.7 In Class
3-13 p376-380
HW 14-38 evens, 50-52, 60, 61
Due TomorrowTest Monday
November 12
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