Unit 6
Quadrilaterals
Lesson 6.1
Properties of Quadrilaterals
Lesson 6.1 Objectives
• Identify a figure to be a quadrilateral.
• Use the sum of the interior angles of a
quadrilateral. (G1.4.1)
Definition of a Quadrilateral
•
A quadrilateral is any four-sided figure with the
following properties:
1.
2.
All sides must be line segments.
Each side must intersect only two other sides.
•
One at each of its endpoints, so that there are no:
i.
ii.
Gaps that do not connect one side to another, or
Tails that extend beyond another side.
Example 6.1
Determine if the figure is a quadrilateral.
1.
5.
Yes
No
Too many intersecting segments
2.
6.
No
Yes
No gaps
3.
Yes
7.
No
Too many sides
4.
No
No curves
8.
No tails
No
Interior Angles
• Recall that the interior angles of any figure
are located in the interior and are formed by
the sides of the figure itself.
180o
Review: What is the sum of
the interior angles of any
triangle?

180o

Review: How many
degrees does a straight
line measure?
???
Review:
What do you
think the sum of
the interior angles
of a quadrilateral
might be?
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
Example 6.2
Find the missing angle.
1.
3.
95o  85o  120o  xo  360o
270o  xo  360o
x  90o
300o  xo  360o
x  60o
2.
4.
262o  xo  360o
x  98o
253o  xo  360o
x  107o
Example 6.3
Find the x.
1.
2.
3.
95o  xo  35o  xo  360o
86  94  (11x  2)  94  360
o
98o  90o  (17 x  3)o  90o  360o
275o  17 x  360o
17 x  85o
x5
o
o
272o  11x  360o
11x  88o
x8
o
o
130o  2 x  360o
2 x  230o
x  115
Lesson 6.1 Homework
• Lesson 6.1 – Properties of Quadrilaterals
• Due Tomorrow
Lesson 6.2
Day 1:
Parallelograms
Lesson 6.2 Objectives
•
•
•
•
Define a parallelogram
Define special parallelograms
Identify properties of parallelograms (G1.4.3)
Use properties of parallelograms to
determine unknown quantities of the
parallelogram (G1.4.4)
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.
– The converse is also true!
• Theorem 6.6
Theorem 6.3:
Opposite Angles of a Parallelogram
• If a quadrilateral is a parallelogram, then its opposite
angles are congruent.
– The converse is also true!
• Theorem 6.7
Example 6.4
Find the missing variables in the parallelograms.
1.
2.
3.


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.
– The converse is also true!
• Theorem 6.8
Q
P
R
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.
• And again, the converse is also true!
– Theorem 6.9
Example 6.5
Find the indicated measure in  HIJK
a)
HI
a)
16
a)
b)
Theorem 6.2
GH
b)
8
b)
c)
Theorem 6.5
KH
c)
10
c)
d)
Theorem 6.2
HJ
d)
16
d)
Theorem 6.5 & 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
Theorem 6.10:
Congruent Sides of a Parallelogram
• If a quadrilateral has one pair of opposite sides that are
both congruent and parallel, then it is a parallelogram.
Example 6.6
Is there enough information to prove the quadrilaterals to be a parallelogram.
If so, explain.
1.
2.
3.
Yes!
Yes!
One pair of parallel and
congruent sides.
(Theorem 6.10)
Both pairs of opposite sides
are congruent.
(Theorem 6.6)
4.
5.
The diagonals bisect
each other.
(Theorem 6.9)
Both pairs of opposite
angles are congruent.
(Theorem 6.7)
6.
Yes!
Yes!
Yes!
Yes!
OR
Both pairs of
Both pairs of opposite One pair of parallel and opposite angles are
congruent sides.
sides are congruent.
congruent.
(Theorem 6.10)
(Theorem 6.6)
(Theorem 6.7)
All consecutive
angles are
supplementary.
(Theorem 6.8)
Lesson 6.2a Homework
• Lesson 6.2: Day 1 – Parallelograms
• Due Tomorrow
Lesson 6.2
Day 2:
(Special) Parallelograms
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 its 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 6.7
Classify the parallelogram.
Explain your reasoning.
2.
1.
Must be
supplementary
Rhombus
Diagonals are perpendicular.
Theorem 6.11
3.



Square
Square Corollary
Rectangle
Diagonals are congruent.
Theorem 6.13
Lesson 6.2b Homework
• Lesson 6.2: Day 2 – Parallelograms
• Due Tomorrow
Lesson 6.3
Trapezoids
and
Kites
Lesson 6.3 Objectives
• Identify properties of a trapezoid. (G1.4.1)
• Recognize an isosceles trapezoid. (G1.4.1)
• Utilize the midsegment of a trapezoid to
calculate other quantities from the
trapezoid.
• Identify a kite. (G1.4.1)
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.
Example 6.8
Find the indicated angle measure of the trapezoid.
1.
2.
CIA
Consecutive
Interior
Angles are
supplementary!
CIA
Recall that a trapezoid has
one set of parallel bases.
xo  62o  180o
x  118o
xo  96o  180o
x  84o
Example 6.9
Consecutive
Interior
Angles are
supplementary!
Find x in the trapezoid.
1.
2.
CIA
CIA
( x  35)o  135o  180o
(5x  25)o  115o  180o
x  170o  180o
5x  140o  180o
x  10
x8
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 6.10
Find the measures of the other three angles.
1.
127o


53o
127o
2.
83o


97o
83o
Supplementary
because of CIA
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
1. parallel to each base and
2. its length is one half the sum of the lengths of the
bases.
•
C
It is the average of the base lengths!
D
AB  CD
MN 
2
N
M
A
B
Example 6.11
Find the indicated length of the trapezoid.
1.
2.
3.
?
?
?
x
7  13
2
x
20
2
x  10
9  12
2
21
x
2
x
x  10.5
32 
43  x
2
Multiply both sides by 2.
Or essentially double the
midsegment!
64  43  x
x  21
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 6.12
Find the missing angle measures.
1.
2.
125o
64o

125o
88o
K = 88
88 + 120 + 88 + J = 360
296 + J = 360
J = 64
60 + K + 50 + M = 360
60 + M + 50 + M = 360
110 + 2M = 360
2M = 250
M = 125
K = 125
But K  M
Example 6.13
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
13
Cause the diagonals are perpendicular!!
a2 + b2 = c2
52 + 122 = c2
25 + 144 = c2
c = 7.07
Use Pythagorean Theorem!
a2 + b2 = c2
13
169 = c2
c = 13
Lesson 6.3 Homework
• Lesson 6.3 – Trapezoids and Kites
• Due Tomorrow
Lesson 6.4
Perimeter and Area
of
Quadrilaterals
Lesson 6.4 Objectives
• Find the perimeter of any type of
quadrilateral. (G1.4.1)
• Find the area of any type of quadrilateral.
(G1.4.3)
Postulate 22:
Area of a Square Postulate
• The area of a square is the square of the
length of its side.
– A = s2
s
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
Example 6.14
Find the perimeter and area of the given quadrilateral.
1.
2.
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 6.23:
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 (b1+b2) h
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 d 1d 2
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.
– A = ½ d 1d 2
d1
d2
Example 6.15
Find the perimeter and area of the given quadrilateral.
1.
2.
3.
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.
Example 6.16
Find the perimeter and area of the given figure.
Assume all corners form a right angle.
Lesson 6.4 Homework
• Lesson 6.4 – Perimeter and Area of Quadrilaterals
• Due Tomorrow
Lesson 6.5
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
• HW
• p367-370
– 8-35, 55-65
• Due Tomorrow
• Test Friday
– March 26