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Quadrilaterals and Polygons
Polygon: A plane figure that is formed by three or more segments (no two of which are
collinear), and each segment (side) intersects at exactly two other sides – one at each
endpoint (Vertex).
Which of the following diagrams are polygons?
Polygons are Named & Classified by the Number of Sides They Have
# of Sides
Type of Polygon
# of Sides Type of Polygon
3
Triangle
8
Octagon
4
Quadrilateral
9
Nonagon
5
Pentagon
10
Decagon
12
Dodagon
#
N-gon
6
7
Hexagon
Heptagon
What type of polygons are the following?
Convex and Concave Polygons
Convex – A polygon is convex if no line that contains a side of the polygon contains a
point in the interior of the polygon.
Concave – A polygon that is not convex
Interior
Interior
Equilateral, Equiangular, and Regular
Diagonals and Interior Angles of a Quadrilateral
Diagonal – a segment that connects to non-consecutive vertices.
Theorem 6.1 – Interior Angles of a Quadrilateral Theorem
The sum of the measures of the interior angles of a quadrilateral is 360O
m<1 + m<2 + m<3 + m<4 = 360o
80o
xo
70o
2xo
Properties of Parallelograms
Theorem 6.2
Q
R
If a quadrilateral is a parallelogram, then
its opposite
_ are_congruent.
_ _ sides
PQ ~
= RS and SP ~
= QR
P
Theorem 6.3
S
Q
R
If a quadrilateral is a parallelogram, then it
opposite angles are congruent.
<P ~
= < R and < Q ~
=<S
P
Theorem 6.4
S
Q
If a quadrilateral is a parallelogram, then
its consecutive angles are supplementary.
R
m<P + m<Q = 180o, m<Q + m<R = 180o
m<R + m<S = 180o, m<S + m<P = 180o
P
S
Q
Theorem 6.5
R
If a quadrilateral is a parallelogram, then
its diagonals
_ _ bisect
_ each
_ other.
QM ~
= SM and PM ~
= RM
P
S
Using the Properties of Parallelograms
F
FGHJ is a parallelogram.
Find the length of:
a. JH
b. JK
5
K
G
3
J
H
Q
PQRS is a parallelogram.
Find the angle measures:
a. m<R
b. m<Q
R
70o
P
S
P
Q
PQRS is a parallelogram.
Find the value of x
3xo
S
120o
R
Proofs Involving Parallelograms
A
2
Given: ABCD and AEFG are
parallelograms
D
G
Statements
1.
2.
3.
4.
ABCD & AEFG are Parallelograms
~<2
<1 =
<2 ~
= <3
~ <3
<1 =
B
1
~
Prove: <1 = < 3
E
C
3
F
Reasons
1. Given
2. Opposite Angles are congruent (6.3)
3. Opposite Angles are Congruent (6.3)
4. Transitive Property of Congruence
Plan: Show that both angles are congruent to <2
Proving Theorem 6.2
A
B
Given: ABCD is a parallelogram
_
~
_
_
~
_
Prove: AB = CD, AD = CB
Statements
1. ABCD is a parallelogram
__
2. Draw Diagonal BD
__
__
__
__
3. AB || CD, and AD || CB
~
4. <ABD = < CDB
~
5. <ADB
= < CBD
__
__
~
6. DB = DB
7. /\__ADB__~= /\ __
CBD __
~
~ CB
8. AB = CD, AD =
D
C
Reasons
1. Given
2. Through any two points there
exists exactly one line
3. Def. of a parallelogram
4. Alternate Interior Angles Theorem
5. Alternate Interior Angles Theorem
6. Reflexive Property of Congruence
7. ASA Congruence Postulate
8. CPCTC
Plan: Insert a diagonal which will allow us to divide the parallelogram into two triangles
Proving Quadrilaterals are Parallelograms
Q
R
Theorem 6.6
If both pairs of opposite sides of a quadrilateral
are congruent, then the quadrilateral is a
parallelogram
P
S
Q
Theorem 6.7
R
If both pairs of opposite angles of a quadrilateral
are congruent, then the quadrilateral is a
parallelogram
P
S
Q
(180-x)o
Theorem 6.8
xo
R
If an angle of a quadrilateral is supplementary to both
of its consecutive angles, then the quadrilateral is a
parallelogram
P xo
S
Q
R
Theorem 6.9
If the diagonals of a quadrilateral bisect each other,
then the quadrilateral is a parallelogram
P
S
Concept Summary – Proving Quadrilaterals are Parallelograms
• Show that both pairs of opposite sides are congruent
• Show that both pairs of opposite sides are parallel
•Show that both pairs of opposite angles are congruent
• Show that one angle is supplementary to BOTH consecutive interior <‘s
• Show that the diagonals bisect each other
• Show that one pair of opposite sides are both congruent and ||
Proving Quadrilaterals are Parallelograms – Coordinate Geometry
How can we prove that the Quad is
a parallelogram?
1. Slope - Opposite Sides ||
C(6,5)
B(1,3)
2. Length (Distance Formula) –
Opposite sides same length
D (7,1)
3. Combination – Show One pair
of opposite sides both || and
congruent
A(2, -1)
Rhombuses, Rectangles, and Squares
Rhombus – a
parallelogram with
four congruent sides
Rectangle – a
parallelogram with
four right angles
Square – a
parallelogram with
four congruent
sides and four right
angles
Parallelograms
Rhombuses
Rectangles
Squares
Using Properties of Special Triangles
A
B
C
D
If ABCD is a rectangle, what else do you know about ABCD?
Corollaries about Special Quadrilaterals
Rhombus Corollary – A quad is a rhombus if and only if it has four congruent sides
Rectangle Corollary – A quad is a rectangle if and only if it has four right angles
Square Corollary – A quad is a square if and only if it is a rhombus and a rectangle
How can we use these special properties and corollaries of a Rhombus?
P
Q
2y + 3
S
5y - 6
R
Using Diagonals of Special Parallelograms
Theorem 6.11:
A parallelogram is a rhombus if and only if its diagonals are perpendicular
Theorem 6.12:
A parallelogram is a rhombus if and only if each diagonal bisects a pair of
opposite angles.
Theorem 6.13:
A parallelogram is a rectangle if and only if its diagonals are congruent
Decide if the statement is sometimes, always, or never true.
1. A rhombus is equilateral. Always
2. The diagonals of a rectangle are _|_. Sometimes
3. The opposite angles of a rhombus are supplementary. Sometimes
4. A square is a rectangle. Always
5. The diagonals of a rectangle bisect each other. Always
6. The consecutive angles of a square are supplementary.
Always
A
Quadrilateral ABCD is Rhombus.
E
7. If m <BAE = 32o, find m<ECD. 32o
8. If m<EDC = 43o, find m<CBA. 86o
9. If m<EAB = 57o, find m<ADC. 66o
D
C
o,
o
10. If m<BEC = (3x -15) solve for x. 35
11. If m<ADE = ((5x – 8)o and m<CBE = (3x +24)o, solve for x 16
12. If m<BAD = (4x + 14)o and m<ABC = (2x + 10)o, solve for x. 26
B
Coordinate Proofs Using the Properties of
Rhombuses, Rectangles and Squares
Using the distance formula and slope, how can we determine the specific
shape of a parallelogram?
Rhombus –
Rectangle –
Square Based on the following Coordinate values, determine if each parallelogram
is a rhombus, a rectangle, or square.
P (-2, 3)
Q(-2, -4)
R(2, -4)
S(2, 3)
P(-4, 0)
Q(3, 7)
R(6, 4)
S(-1, -3)
H
I
O
Given: HIJK is a parallelogram
/\ HOI ~
= /\ JOI
Prove: HIJK is a Rhombus
Statements
K
J
Reasons
R
Given: RECT is a Rectangle
E
A
~
Prove: /\ ART = /\ ACE
T
Statements
C
Reasons
P
Q
Given: PQRT is a Rhombus
Prove: PR bisects <TPQ and < QRT,
and QT bisects <PTR and <PQP
Statements
Plan:
T
R
Reasons
First prove that Triangle PRQ is congruent to Triangle PRT; and Triangle
TPQ is congruent to Triangle TRQ
Trapezoids and Kites
A Trapezoid is a Quad with exactly one
pair of parallel sides. The parallel sides
are the BASES. A Trapezoid has exactly
two pairs of BASE ANGLES
In trapezoid ABCD, Which 2 sides are the
bases? The legs? Name the pairs of
base angles.
A
>
B
>
D
A
C
>
B
If the legs of the trapezoid are congruent,
then the trapezoid is an Isosceles Trapezoid.
D
>
C
Theorems of Trapezoids
A
B
>
Theorem 6.14
If a trapezoid is isosceles, then each pair of base angles is
congruent.
<A ~
= <B ~
= <C ~
= <D
>
D
A
Theorem 6.15
>
C
B
If a trapezoid has a pair of congruent base angles,
then it is an isosceles trapezoid.
ABCD is an isosceles trapezoid
>
D
A
Theorem 6.16
C
>
B
A trapezoid is isosceles if and only if its diagonals are
congruent.
_ _
~ BD
ABCD is isosceles if and only if AC =
D
>
C
Kites and Theorems about Kites
A kite is a quadrilateral that has two
pairs of consecutive congruent sides,
But opposite sides are NOT
congruent.
Theorem 6.18
If a Quad is a Kite, then its diagonals are
perpendicular.
Theorem 6.19
If a Quad is a kite then exactly one pair of
opposite angles are congruent
Using the Properties of a Kite
X
Find the length of WX, XY, YZ, and WZ.
12
20
U
W
12
Y
12
Z
Find the angle measures of <HJK and < HGK
J
H 132o
60o
G
K
Summarizing the Properties of Quadrilaterals
Quadrilaterals
Kites
______________
Parallelograms
_________________
Trapezoids
________________
Rhombus
Squares
Rectangles
____________ _____________ ____________
Isosceles Trap.
______________
Properties of Quadrilaterals
Property
Both pairs of Opp.
sides a ||
Exactly one pair of
Opp. Sides are ||
Diagonals are _|_
Diagonals are =
Diagonals Bisect
each other
Both pairs of Opp.
Sides are =
Exactly one pair of
opp. Sides are =
All Sides are =
Both pairs of Opp.
<'s are =
Exactly one pair of
Opp <'s are =
All <'s are =
Rectangle
X
X
Rhombus
Square
X
Kite
Trapezoid
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Using Area Formulas
Area of a Square Postulate
The area of a square is the square of the length of its side.
Area Congruence Postulate
If two polygons are congruent then they have the same area.
Area Addition Postulate
The area of a region is the sum of the area of its non-overlapping sides.
Area of a Rectangle
h
The area of a rectangle is the product of its base and height.
A = bh
b
Area of a Parallelogram
The area of a parallelogram is the product of a base, and it’s
corresponding height
h
A = bh
b
Area of a Triangle
The area of a triangle is one half the product of a base and its
corresponding height
h
A = ½bh
b
Q
R
Given: /\ RQP ~
= /\ ONP
__
R is the midpoint of MQ
Prove: MRON is a parallelogram
P
M
Statements
~
1. /\ RQP = /\ ONP
R is the midpoint of MQ
__
__
~
2. MR = RQ
__ __
3. RQ ~= NO
__
__
~
4. MR = NO
Reasons
1. Given
5. <QRP ~
= < NOP
__ __
6. MQ || NO
5. CPCTC
7. MRON is a parallelogram
7. Theorem 6.10
O
N
2. Definition of a midpoint
3. CPCTC
4. Transitive Property of Congruency
6. Alternate Interior <‘s Converse
U
V
2
W
3 4
1
Given: UWXZ is a parallelogram, <1 ~
= <8
Prove: UVXY is a parallelogram
8
5
Z
Statements
1. UWXZ is a parallelogram
__
Y
7
X
Reasons
1. Given
__
2. UW || ZX
__
6
2. Definition of a parallelogram
__
3. UV || YX
3. Segments of Congruent Segments
4. <Z ~
= <W
4. Opposite <‘s of a parallelogram are =
5. <1 ~
= <8
5. Given
6. <5 ~
= <4
6. Third Angles Theorem
6. <4 ~
= <7
7. Alternate Interior Angles Theorem
6. <5 ~
= <7
8. Transitive Property of Congruence
__
__
7. UY || VX
9. Corresponding Angles Converse
8. UVXY is a parallelogram
10. Definition of a Parallelogram
L
K
J
M
Given: GIJL is a parallelogram
Prove: HIKL is a parallelogram
G
Statements
1. GIJL is a parallelogram
__
H
I
Reasons
1. Given
__
2. GI || LJ
2. Definition of a parallelogram
~
3. <GIL = <JLI
3. Alternate Interior Angles Theorem
4. GJ Bisects LI
4. Diagonals of a parallelogram bisect
__ ~ __
5. MI = ML
5. Definition of a Segment Bisector
6. <HMI ~
= <KML
6. Vertical Angles Theorem
~ /\ KML
7. /\ HMI =
7. ASA Congruence Postulate
__
__
8. MH ~
= MK
8. CPCTC
9. HK and IL Bisect Each other
9. Definition of a Segment Bisector
10. HIKL is a parallelogram
10. Theorem 6.9