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Good Morning
Unit 6
Quadrilaterals
Part 1
Parallelograms
Definition
• A parallelogram is a quadrilateral whose
opposite sides are parallel.
B
C
D
A
• Its symbol is a small figure:
AB CD and BC AD
Naming a Parallelogram
• 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.
B
A
C
D
Basic Properties
• There are four basic properties of all
parallelograms.
• These properties have to do with the angles,
the sides and the diagonals.
Opposite Sides
Theorem Opposite sides of a parallelogram
are congruent.
B
C
A
D
• That means that AB  CD and BC
.  AD
• So, if AB = 7, then _____ = 7?
Opposite Angles
• One pair of opposite angles is A and
 C. The other pair is  B and  D.
B
A
C
D
Opposite Angles
Theorem Opposite angles of a
parallelogram are congruent.
• Complete: If m  A = 75 and
m
 B = 105, then m  C = ______
and m  D = ______ .
B
A
C
D
Consecutive Angles
• Each angle is consecutive to two other
angles. A is consecutive with  B and
 D.
B
A
C
D
Consecutive Angles in Parallelograms
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 = _____?
B
A
C
D
Consecutive
INTERIOR
Angles are
Supplementary!
Diagonals
• Diagonals are segments that join nonconsecutive vertices.
• For example, in this diagram, the only two
diagonals are
.
AC and BD
B
A
C
D
Diagonal Property
When the diagonals of a parallelogram intersect, they
meet at the midpoint of each diagonal.
• So, P is the midpoint of AC and BD.
• Therefore, they bisect each other;
so AP  PC and BP  PD .
• But, the diagonals are not congruent!
AC  BD
B
C
P
A
D
Diagonal Property
Theorem The diagonals of a parallelogram bisect each
other.
B
C
P
A
D
Parallelogram Summary
• 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.
Examples
• 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 =______ .
Examples (cont’d)
•
•
•
•
•
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 = _____ .
Part 2
Tests for
Parallelograms
Review: Properties of
Parallelograms
•
•
•
•
•
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
The diagonals bisect each other.
How can you tell if a quadrilateral
is a parallelogram?
• 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.
Proving Quadrilaterals as Parallelograms
Theorem 1: If both pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a parallelogram .
If EF  GH; FG  EH, then Quad. EFGH is a parallelogram.
H
G
Theorem 2:
E
F
If one pair of opposite sides of a quadrilateral are both congruent and
parallel, then the quadrilateral is a parallelogram .
If EF  GH and EF || HG, then Quad. EFGH is a parallelogram.
Theorem:
Theorem 3:
If both pairs of opposite angles of a quadrilateral are congruent, then
the quadrilateral is a parallelogram.
G
H
If H  F and E  G,
then Quad. EFGH is a parallelogram.
M
Theorem 4:
E
F
If the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram . If M is the midpo int of EG and FH
then Quad. EFGH is a parallelogram.
EM = GM and HM = FM
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 .
Examples ……
Example 1: Find the values of x and y that ensures the quadrilateral
y+2
is a parallelogram.
6x = 4x + 8
2y = y + 2
6x
4x+8
2x = 8
y=2
2y
x=4
Example 2: Find the value of x and y that ensure the quadrilateral is
a parallelogram. 2x + 8 = 120
5y + 120 = 180
(2x + 8)° 5y°
120°
2x = 112
5y = 60
x = 56
y = 12
Part 3
Rectangles
Lesson 6-3: Rectangles
24
Rectangles
Definition: A rectangle is a quadrilateral with four right angles.
Is a rectangle is a parallelogram?
Yes, since opposite angles are congruent.
Thus a rectangle has all the properties of a parallelogram.
•
•
•
•
•
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
Lesson 6-3: Rectangles
25
Properties of Rectangles
Theorem: If a parallelogram is a rectangle, then its diagonals
are congruent.
Therefore, ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles.
A
B
E
D
C
Converse: If the diagonals of a parallelogram are congruent ,
then the parallelogram is a rectangle.
Lesson 6-3: Rectangles
26
Properties of Rectangles
Parallelogram Properties:

Opposite sides are parallel.

Opposite sides are congruent.
A

Opposite angles are congruent.

Consecutive angles are supplementary.

Diagonals bisect each other.
Plus:
D

All angles are right angles.

Diagonals are congruent.

B
E
C
Also: ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles
triangles
Lesson 6-3: Rectangles
27
Examples…….
1. If AE = 3x +2 and BE = 29, find the value of x.
x = 9 units
10.5 units
2. If AC = 21, then BE = _______.
3. If m<1 = 4x and m<4 = 2x, find the value of x.
x = 18 units
4. If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6.
A
B
2
m<1=50,
m<3=40,
m<4=80,
m<5=100,
m<6=40
3
1
4
E
5
6
D
Lesson 6-3: Rectangles
C
28
Part 4
Rhombi
and
Squares
Lesson 6-4: Rhombus & Square
29
Rhombus
Definition: A rhombus is a quadrilateral with four congruent sides.
Is a rhombus a parallelogram?
Yes, since opposite sides are congruent.
Since a rhombus is a parallelogram the following are true:
• Opposite sides are parallel.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• Diagonals bisect each other.
Lesson 6-4: Rhombus & Square
30
Rhombus
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.
Lesson 6-4: Rhombus & Square
31
Properties of a Rhombus
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!
Lesson 6-4: Rhombus & Square
32
Properties of a Rhombus
Since a rhombus
.
is a parallelogram the following are true:
• 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!
Lesson 6-4: Rhombus & Square
33
Rhombus Examples .....
Given: ABCD is a rhombus. Complete the following.
A
1.
2.
3.
9 units
If AB = 9, then AD = ______.
B
1 2
3
65°
If m<1 = 65, the m<2 = _____.
4
m<3 = ______.
90°
5
D
4.
100°
If m<ADC = 80, the m<DAB = ______.
5.
10
If m<1 = 3x -7 and m<2 = 2x +3, then x = _____.
Lesson 6-4: Rhombus & Square
E
C
34
Square
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.
• 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.
35
Squares – Examples…...
Given: ABCD is a square. Complete the following.
1.
10 unitsand DC = _____.
10 units
If AB = 10, then AD = _____
A
2.
5 units
If CE = 5, then DE = _____.
B
1 2
3
4
E
3.
90°
m<ABC = _____.
8 7
4.
45°
m<ACD = _____.
5.
90°
m<AED = _____.
D
Lesson 6-4: Rhombus & Square
6
5
C
36
Part 5
Trapezoids
and Kites
Lesson 6-5: Trapezoid & Kites
37
Trapezoid
Definition: A quadrilateral with exactly one pair of parallel sides.
The parallel sides are called bases and the non-parallel sides are
called legs.
Base
Leg
Trapezoid
Leg
Base
Lesson 6-5: Trapezoid & Kites
38
Median of a Trapezoid
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.
b
1
1
median  (b1  b2 )
2
Median
b2
Lesson 6-5: Trapezoid & Kites
39
Isosceles Trapezoid
Definition: A trapezoid with congruent legs.
Isosceles
trapezoid
Lesson 6-5: Trapezoid & Kites
40
Properties of Isosceles Trapezoid
1. Both pairs of base angles of an isosceles trapezoid are congruent.
A  B and D  C
2. The diagonals of an isosceles trapezoid are congruent.
AC  DB
B
A
D
Lesson 6-5: Trapezoid & Kites
C
41
Kite
Definition: A quadrilateral with two distinct pairs of congruent
adjacent sides.
Theorem:
Diagonals of a kite are
perpendicular.
Lesson 6-5: Trapezoid & Kites
42
Flow Chart
Quadrilaterals
Kite
Parallelogram
Trapezoid
Rhombus
Rectangle
Isosceles
Trapezoid
Square
Lesson 6-5: Trapezoid & Kites
43