Parallelograms Parallelograms Parallelograms Parallelograms

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Parallelograms
Quadrilaterals

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A parallelogram is a quadrilateral whose opposite
sides are parallel.
Its symbol is a small figure:
Definition
AB CD and BC AD

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A parallelogram is named using all four vertices.
You can start from any one vertex, but you must
continue in a clockwise or counterclockwise
direction.
For example, this can be either
ABCD or ADCB.
Naming a Parallelogram

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There are four basic properties of all
parallelograms.
These properties have to do with the angles,
the sides and the diagonals.
Basic Properties

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Theorem Opposite sides of a parallelogram are
congruent.
That means that .
So, if AB = 7, then ______ = 7?
Opposite Sides
AB  CD and BC  AD

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One pair of opposite angles is A and
 C. The other pair is  B and  D.
Opposite Angles

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Theorem Opposite angles of a parallelogram
are congruent.
Complete: If m  A = 75 and m  B =
105, then m  C = ______ and m  D = ______ .
Opposite Angles

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Each angle is consecutive to two other angles.
A is consecutive with  B and  D.
Consecutive Angles

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Theorem: Consecutive angles in a parallelogram are
supplementary.
Therefore, m  A + m  B = 180 and m  A + m  D =
180.
If m<C = 46, then m  B = _____?
Consecutive Angles in Parallelograms
Consecutive
INTERIOR
Angles are
Supplementary!

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Diagonals are segments that join non-consecutive
vertices.
For example, in this diagram, the only two diagonals
are .
Diagonals
AC and BD

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When the diagonals of a parallelogram intersect, they meet at
the midpoint of each diagonal.
So, P is the midpoint of .
Therefore, they bisect each other; so and .
But, the diagonals are not congruent!
Diagonal Property
AC and BD
AP  PC BP PD
AC  BD

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Theorem: The diagonals of a parallelogram bisect
each other.
Diagonal Property

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By its definition, opposite sides are
parallel.
Other properties (theorems):
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
The diagonals bisect each other.
Parallelogram Summary

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1. Draw HKLP.
2. Complete: HK = _______ and HP = ________ .
3. m<K = m<______ .
4. m<L + m<______ = 180.
5. If m<P = 65, then m<H = ____, m<K =
______ and m<L =______ .
Activity 2

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6. Draw in the diagonals. They intersect at M.
7. Complete: If HM = 5, then ML = ____ .
8. If KM = 7, then KP = ____ .
9. If HL = 15, then ML = ____ .
10. If m<HPK = 36, then m<PKL = _____ .
Activity 2 (cont’d)

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Tests for
Parallelograms
Part 2

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Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
The diagonals bisect each other.
Review: Properties of
Parallelograms

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• Defn: A quadrilateral is a parallelogram iff opposite
sides are parallel.
• Property: If a quadrilateral is a parallelogram, then
opposite sides are parallel.
• Test: If opposite sides of a quadrilateral are parallel,
then it is a parallelogram.
How can you tell if a quadrilateral is a
parallelogram?

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Proving Quadrilaterals as Parallelograms
If both pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a parallelogram .
Theorem 1:
H G
E F
If one pair of opposite sides of a quadrilateral are both congruent and
parallel, then the quadrilateral is a parallelogram .
Theorem 2:
If EF GH; FG EH, then Quad. EFGH is a parallelogram.
 
If EF GH and EF || HG, then Quad. EFGH is a parallelogram.


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Proving Quadrilaterals as Parallelograms
If both pairs of opposite angles of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorem 3:
If the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram .
Theorem 4:
H G
E
F
M
,
If H F and E G
     
int
If M is themidpo of EG and FH
then Quad. EFGH is a
parallelogram.
then Quad. EFGH is a
parallelogram.
EM = GM and HM = FM

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5 ways to prove that a quadrilateral is a
parallelogram.
1. Show that both pairs of opposite sides are || . [definition]
2. Show that both pairs of opposite sides are  .
3. Show that one pair of opposite sides are both || and  .
4. Show that both pairs of opposite angles are  .
5. Show that the diagonals bisect each other .

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Examples ……
Find the values of x and y that ensures the
quadrilateral is a parallelogram.
Example 1:
6x
4x+8
y+2
2y
6x = 4x + 8
2x = 8
x = 4
2y = y + 2
y = 2
Example 2: Find the value of x and y that ensure the quadrilateral is a
parallelogram.
120°
5y°
(2x + 8)° 2x + 8 =
120
2x = 112
x = 56
5y + 120 =
180
5y = 60
y = 12

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Rectangles
Part 3
Lesson 6-3: Rectangles
23

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Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
Rectangles
Lesson 6-3: Rectangles
24
Definition: A rectangle is a quadrilateral with four right angles.
Is a rectangle a parallelogram?
Thus a rectangle has all the properties of a parallelogram.
Yes, since opposite angles are congruent.

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Properties of Rectangles
Lesson 6-3: Rectangles
25
Therefore, ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles
triangles.
If a parallelogram is a rectangle, then its diagonals
are congruent.
E
D C
B
A
Theorem:
Converse: If the diagonals of a parallelogram are congruent , then the
parallelogram is a rectangle.

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Properties of Rectangles
Lesson 6-3: Rectangles
26
Parallelogram Properties:
 Opposite sides are parallel.
 Opposite sides are congruent.
 Opposite angles are congruent.
 Consecutive angles are supplementary.
 Diagonals bisect each other.
Plus:
 All angles are right angles.
 Diagonals are congruent.
 Also: ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles
E
D C
B
A

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Examples…
Lesson 6-3: Rectangles
27
1. If AE = 3x +2 and BE = 29, find the value of x.
2. If AC = 21, then BE = _______.
3. If m<1 = 4x and m<4 = 2x, find the value of x.
4. If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6.
m<1=50,
m<3=40,
m<4=80,
m<5=100,
m<6=40
10.5 units
x = 9 units
x = 18 units
6
5
4
3
2
1
E
D C
B
A

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Rhombi
and
Squares
Part 4
Lesson 6-4: Rhombus & Square
28

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Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
Rhombus
Lesson 6-4: Rhombus & Square
29
Definition: A rhombus is a quadrilateral with four congruent
sides.
Since a rhombus is a parallelogram the following are true:
Is a rhombus a parallelogram?
Yes, since opposite sides are congruent.

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Rhombus
Lesson 6-4: Rhombus & Square
30
Note: The four small triangles are congruent, by SSS.
This means the diagonals form
four angles that are congruent,
and must measure 90 degrees
each.
So the diagonals are perpendicular.
This also means the diagonals
bisect each of the four angles
of the rhombus
So the diagonals bisect opposite angles.

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Properties of a Rhombus
L
e
s
s
o
n
6
-
4
:
R
h
o
m
b
u
s
&
S
q
u
a
r
e
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Theorem: The diagonals of a rhombus are perpendicular.
Theorem: Each diagonal of a rhombus bisects a pair of opposite angles.
Note: The small triangles are RIGHT and CONGRUENT!

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• Opposite sides are parallel.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• Diagonals bisect each other.
Plus:
• All four sides are congruent.
• Diagonals are perpendicular.
• Diagonals bisect opposite angles.
• Also remember: the small triangles are RIGHT and
CONGRUENT!
Properties of a Rhombus
Lesson 6-4: Rhombus & Square
32
.
Since a rhombus is a parallelogram the following are true:

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Given: ABCD is a rhombus. Complete the following.
1. If AB = 9, then AD = ______.
2. If m<1 = 65, the m<2 = _____.
3. m<3 = ______.
4. If m<ADC = 80, the m<DAB = ______.
5. If m<1 = 3x -7 and m<2 = 2x +3, then x = _____.
Rhombus Examples .....
Lesson 6-4: Rhombus & Square
33
9 units
65°
90°
100°
10

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• Opposite sides are parallel.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• Diagonals bisect each other.
Plus:
• Four right angles.
• Four congruent sides.
• Diagonals are congruent.
• Diagonals are perpendicular.
• Diagonals bisect opposite angles.
Square
34
Definition: A square is a quadrilateral with four congruent angles
and four congruent sides.
Since every square is a parallelogram as well as a rhombus and
rectangle, it has all the properties of these quadrilaterals.

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Given: ABCD is a square. Complete the following.
1. If AB = 10, then AD = _____ and DC = _____.
2. If CE = 5, then DE = _____.
3. m<ABC = _____.
4. m<ACD = _____.
5. m<AED = _____.
Squares – Examples…...
Lesson 6-4: Rhombus & Square
35
10 units 10 units
5 units
90°
45°
90°

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Trapezoids
and Kites
Part 5
Lesson 6-5: Trapezoid & Kites
36

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Trapezoid
Lesson 6-5: Trapezoid & Kites
37
A quadrilateral with exactly one pair of parallel sides.
Definition:
Base
Leg Trapezoid
The parallel sides are called bases and the non-parallel sides are called
legs.
Leg
Base

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Lesson 6-5: Trapezoid & Kites
38
The median of a trapezoid is the segment that joins the midpoints of
the legs. (It is sometimes called a midsegment.)
• Theorem - The median of a trapezoid is parallel to the bases.
• Theorem - The length of the median is one-half the sum of the
lengths of the bases.
Median
1
b
2
b
1 2
1
( )
2
median b b
 
Median of a Trapezoid

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Isosceles Trapezoid
Lesson 6-5: Trapezoid & Kites
39
A trapezoid with congruent legs.
Definition:
Isosceles
trapezoid

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Properties of Isosceles Trapezoid
A B and D C
     

AC DB
Lesson 6-5: Trapezoid & Kites
40
2. The diagonals of an isosceles trapezoid are
congruent.
1. Both pairs of base angles of an isosceles trapezoid are congruent.
A B
C
D

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Kite
Lesson 6-5: Trapezoid & Kites
41
A quadrilateral with two distinct pairs of congruent
adjacent sides.
Definition:
Theorem:
Diagonals of a kite
are
perpendicular.
Theorem:
The Area of a kite is
half the product of
the lengths of its
diagonals.

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Lesson 6-5: Trapezoid & Kites
42
Isosceles
Trapezoid
Quadrilaterals
Rectangle
Parallelogram
Rhombus
Square
Flow Chart
Trapezoid
Kite