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Geometry 1
Unit 6:
Quadrilaterals
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Geometry 1
6.1 Polygons
Polygon
Unit 6: Quadrilaterals
Sides
Vertex
polygons
not polygons
How to name a Polygon:
A
E
B
D
C
3
tri- 3
hepta- 7
quadr- 4
oct- 8
penta- 5
nona- 9
hexa- 6
dec- 10
Use the prefix chart above and the given definition to
write a definition of the boldfaced word.
1. triplet: One of three children born at one birth.
Quadruplet
2. quadrennial: Happening once every four years.
Octennial
3. decathalon: An athletic contest that consists of ten events for
each participant.
Pentathalon
4. hexapod: Having six legs or feet.
Tripod
5. octogenarian: A person between eighty and ninety years of
age.
Nonagenarian
6. pentagram: A five-pointed star.
Hexagram
7. octahedron: A solid geometric figure with eight plane faces.
Decahedron
8. nonagon: A polygon with nine sides and nine angles.
Heptagon
9. heptameter: A line of verse consisting of seven metrical feet.
Pentameter
4
Polygons
Number of Sides
3
4
5
6
7
8
9
Name
10
11
12
n
Diagonals
Convex
polygons
Concave
polygons
Example 1
Identify the polygon and state whether it is convex or concave.
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Can a polygon be equiangular and not equilateral?
Draw an example.
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Equilateral
Polygon
Equiangular
Polygon
Regular
Polygon
Example 2
Decide whether the polygon is regular.
Interior Angles of a Quadrilateral Theorem
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1
__________________________________________
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3
7
8
H
Example 3
Find mF, mG, and mH.
G
x
E
55
°
x
F
Example 4
Use the information in the diagram to solve for x
100° 120°
2x + 30
3x – 5
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Geometry 1
6.2 Properties of Parallelograms
Parallelogram
Unit 6: Quadrilaterals
Opposite
Sides of a
Parallelogram
Theorem
Q
P
Opposite
Angles in a
Parallelogram
Theorem
R
S
Q
P
Consecutive
Angles in a
Parallelogram
Theorem
R
S
Add
to
equal
180°
Diagonals in a
Parallelogram
Theorem
Q
R
M
P
Example 1
GHJK is a parallelogram. Find each unknown length
JH
K
LH
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8
G
S
J
L
H
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Example 2
In parallelogram ABCD, mC = 105°. Find the measure of each angle.
mA
mD
Example 3
WXYZ is a parallelogram. Find the value of x.
3x + 18°
4x – 9°
Example 4
Given:: ABCD is a
parallelogram.
Prove: 2  4
Statement
Reasons
ABCD is a
parallelogram
AD || BC
A
B
2
1
4
D
3
2  1
AB || CD
Alternate interior angles
theorem
C
2  4
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Example 5
Given: ACDF is a
parallelogram.
ABDE is a parallelogram.
Prove: ∆BCD  ∆EFA
A
B
Statement
Reason
ACDF is a
parallelogram.
ABDE is a
parallelogram.
C
Opposite sides of
a parallelogram
are congruent
AC = DF
AB = DE
AC = AB + BC
F
E
D
DF = DE + EF
AC = DE + DF
AB + BC = AB +
EF
BC = EF
Def of Congruent
∆BCD  ∆EFA
Example 6
A four-sided concrete slab has consecutive angle measures of 85°, 94°, 85°, and 96°.
Is the slab a parallelogram? Explain.
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Investigating Properties of Parallelograms
Cut 4 straws to form two congruent pairs.
Partly unbend two paperclips, link their smaller ends, and insert the larger
ends into two cut straws. Join the rest of the straws to form a quadrilateral
with opposite sides congruent.
Change the angles of your quadrilateral. Is your quadrilateral a
parallelogram?
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Geometry 1
6.3 Proving Quadrilaterals are Parallelograms
Unit 6: Quadrilaterals
Converse of
the Opposite
Sides of a
Parallelogram
Theorem
A
B
D
Converse of
the Opposite
Angles in a
Parallelogram
Theorem
Converse of
the
Consecutive
Angles in a
Parallelogram
Theorem
C
A
B
D
C
A
x°
(180
–
x)°
x°
B
C
D
Converse of
the Diagonals
in a
Parallelogram
Theorem
A
B
M
D
C
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Example 1
Given: ∆PQT  ∆RST
Prove: PQRS is a parallelogram.
P
Q
Statements
Reasons
∆PQT  ∆RST
CPCTC
PT = RT
ST = QT
Def. of bisect
T
S
PQRS is a parallelogram
R
Example 2
A gate is braced as shown. How do you know that opposite sides of the gate are
congruent?
Congruent
and Parallel
Sides
Theorem
B
A
C


D
To determine if a quadrilateral is a parallelogram, you need to know one of the following:
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Example 3
Show that A(-1,2), B(3,2), C(1,-2), and D(-3,-2) are the vertices of a parallelogram.
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Answer Sometimes, Always or Never
1. A square is a rectangle.
2. A rectangle is a square.
3. A rhombus is a rectangle.
4. A square is a rhombus.
5. A rhombus is a rectangle.
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Geometry 1
6.4 Rhombuses, Rectangles, and Squares
Unit 6: Quadrilaterals
Rectangle
Rhombus
Square
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Example 1
Decide if each statement is always, sometimes or never true.
A rhombus is a rectangle
A parallelogram is a rectangle
A rectangle is a square
A square is a rhombus
Example 2
Given FROG is a rectangle, what else do you know about FROG?
F
R
G
Example 3
O
EFGH is a rectangle. K is the midpoint of FH. EG = 8z – 16,
What is the measure of segment EK?
What is the measure of segment GK?
Rhombus
Corollary
Rectangle
Corollary
Square
Corollary
Perpendicular
Diagonals of a
Rhombus
Theorem
B C
  
A


D
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Diagonals
Bisecting
Opposite
Angles
Theorem.
B
C
A
Diagonals in a
Rectangle
Theorem
D
A
B
D
C
Example 4
You cut out a parallelogram shaped quilt piece and measure the diagonals to
be congruent. What is the shape?
An angle formed by the diagonals of the quilt piece measures 90°. Is the
shape a square?
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Geometry 1
6.5 Trapezoids and Kites
Unit 6: Quadrilaterals
Trapezoid
Bases
Pairs of Base
Angles
Legs
Isosceles
Trapezoid
Base Angles
of an
Isosceles
Trapezoid
Theorem
Congruent
Base Angles
in a Trapezoid
Theorem.
Diagonals in
an Isosceles
Trapezoid
Theorem
Midsegment
of a trapezoid
A
B

D
A

C
B

D
A
D

C
B


C
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Midsegment
Theorem for
Trapezoids
B
M
A
Kite


Diagonals of a
Kite Theorem
Opposite
Angles in a
Kite Theorem
Example 4
GHJK is a kite. Find HP.
√29
5
G
H
J
P
K
Example 5
RSTU is a kite. Find mR, mS, and mT.
S
R
x + 30°
x°
T
125°
U
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C
N
D
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Geometry 1
6.6 Special Quadrilaterals
Property
Unit 6: Quadrilaterals
Parallelogram Rectangle Rhombus Square Trapezoid Kite
1.Both pairs of
opposite sides are
congruent
2. Diagonals are
congruent
3. Diagonals are
perpendicular
4. Diagonals
bisect each other
5. Consecutive
angles are
supplementary
6. Both pairs of
opposite angles
are congruent
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Geometry 1
6.7 Areas of Triangles and Quadrilaterals
Unit 6: Quadrilaterals
Example 1
Example 2
What is the base of a triangle that has an area of 48 and a height of 3?
Example 3
A rectangle has an area of 100 square meters and a height of 25 meters. Are all
the rectangles with these dimensions congruent?
Example 4
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Example 5
What is the height of a parallelogram that has an area of 96 square feet and a base
length of 8 feet?
Example 6
Find the area of trapezoid EFGH.
E(-2, 3), F(2, 4), G(2, -2), H(-2, -1)
Example 7
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Example 8
Example 9
Example 10
Find the area of rhombus EFGH if EG = 10 and FH = 15.
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