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Unit 8
“Quadrilaterals”
Academic Geometry
Spring 2014
Name_____________________________ Teacher__________________ Period______
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Unit 8 at a glance
“Quadrilaterals”
This unit focuses on revisiting prior knowledge of polygons and extends to formulate, test,
and apply conjectures about quadrilaterals. Students will be able to identify
quadrilaterals by the given properties and apply the properties to solve both purely
mathematical and real world situations.
Essential Questions
 How are polygons related? What properties do they share?
 What are the differences and similarities between the types of quadrilaterals (square,
rectangle, rhombus, parallelogram, trapezoid, and kite)? And how does knowing this
help me?
In Unit 8, students will…
 Apply the terms convex, concave, n-gon, equilateral, equiangular, and regular to
describe polygons when solving problems involving area, perimeter and circumference
in both real-world and purely mathematical situations;
 Formulate, test, and apply conjectures (based on explorations and concrete models)
involving the sum of the measures of interior and exterior angles of convex polygons
and the measures of each interior and exterior angle of a regular polygon to solve
problems in both real-world and purely mathematical situations;
 Formulate, test, and apply conjectures (based on explorations and concrete models)
involving the properties of parallelogram (including angle and side measure
relationships) to solve problems in both real-world and purely mathematical
situations;
 Formulate, test, and apply conjectures (based on explorations and concrete models)
involving the conditions that ensure a quadrilateral is a parallelogram including
opposite side, opposite angle and diagonal relationships and solve problems in both
real-world and purely mathematical situations requiring axiomatic and coordinate
approaches;
 Formulate, test, and apply conjectures (based on explorations and concrete models)
involving the properties of rhombuses, rectangles and squares to solve problems in
both real-world and purely mathematical situations;
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 Formulate, test, and apply conjectures (based on explorations and concrete models)
involving the properties of trapezoids and kites to solve problems in both real-world
and purely mathematical situations requiring axiomatic or coordinate geometry
approaches;
 Compare and contrast quadrilaterals (parallelograms, rhombuses, rectangles, squares,
trapezoids, kites) and their properties to identify them and to solve problems in both
real-world and purely mathematical situations.
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Vocabulary
concave polygons
convex polygons
diagonal
equiangular
exterior angles
interior angles
isosceles trapezoid
kite
midsegment
n-gons
opposite angle
parallelogram
quadrilateral
rectangle
regular polygons
rhombus
square
trapezoid
vertices
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*Quadrilateral
Meet the Quadrilateral Family
I have exactly four sides.
*Parallelogram
The sum of my interior angles is 3600.
I have:
- 2 sets of parallel sides
- 2 sets of congruent sides
- opposite angles congruent
- consecutive angles supplementary
- diagonals bisect each other
*Rhombus
- diagonals form 2 congruent triangles
I have all of the properties of the parallelogram PLUS
- 4 congruent sides
- diagonals bisect angles
- diagonals perpendicular
*Rectangle
I have all of the properties of the parallelogram PLUS
- 4 right angles
- diagonals congruent
*Square
Hey, look at me!
I have all of the properties of the parallelogram
AND the rectangle AND the rhombus.
I have it all!
*Trapezoid
I have only one set of parallel sides.
*Isosceles Trapezoid
I have:
- only one set of parallel sides
- base angles congruent
- legs congruent
- diagonals congruent
- opposite angles supplementary
*Kite
I have:
- 2 sets of consecutive congruent sides
- only one pair of opposite angles congruent
- diagonals perpendicular
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8.1 NOTES
Essential Vocabulary
Polygon
Not Polygons
Concave
Equilateral
Convex
Equiangular
Regular
Triangle
Octagon
Quadrilateral
Nonagon
Pentagon
Decagon
Hexagon
Dodecagon
Heptagon
n-gon
(an n-sided shape)
ANY shape can be called “n”-gon based on
the number of sides. Polygons with more
than 10 sides are usually referred to as
“n”-gons
Ex. 14-gon, 32-gon, 100-gon
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Interior and Exterior Angles of a Polygon
In a polygon, two vertices that are endpoints of the same side are called consecutive
vertices. A diagonal of a polygon joins two non-consecutive vertices of a polygon.
Consecutive vertices
Choose any vertex and
draw every diagonal
possible from that vertex.
Notice that when you draw all the diagonals of a polygon from one vertex, you divide the
polygon into ___________________________________. Recall that the triangle sum theorem states
_____________________________________________________________________________________________________.
For each polygon in the table, draw all the diagonals from one vertex. Complete the table.
Polygon
# of Sides
# of triangles
formed
Interior Angle
Sum of Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
n-gon
POLYGON INTERIOR ANGLES THEOREM:
For an n-sided convex polygon, the sum of all the interior angles is ____________________.
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The exterior angle sum of a polygon does not depend on the number of sides on the
polygon. To prove this, use the diagrams below to calculate each exterior angle of the
polygons. Remember that an interior angle and its adjacent exterior angle form a linear
pair (so their sum is ___________).
triangle
quadrilateral
85⁰
pentagon
110⁰
130⁰
50⁰
125⁰
115⁰
85⁰
55⁰
90⁰
Interior
angle sum
40⁰
180⁰
95⁰
360⁰
100⁰
540⁰
Exterior
angle sum
POLYGON EXTERIOR ANGLES THEOREM:
The sum of all the exterior angles of a convex polygon (one exterior angle at each
vertex is always ________________________.
Example 1:
Example 2:
Find the sum of the measures of the interior
angles of a convex octagon.
The sum of the measures of the interior angles
of a convex polygon is 1440⁰. How many sides
does the polygon have?
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Example 3:
Example 4:
Find the value of x.
A trampoline is shaped like a regular
dodecagon (12 sides). Find the measure of
each interior and exterior angle.
Example 5:
Find the value of x.
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8.1 HOMEWORK
Find the sum of the measures of the interior angles of the indicated convex polygon.
1.
11-gon
2. 40-gon
The sum of the measures of the interior angles of a convex polygon is given. Classify
the polygon by the number of sides.
3.
4.
5.
Find the value of .
6.
7.
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8.
The measures of the interior angles of a convex octagon are
and
What is the measure of the
smallest interior angle?
Find the measures of an interior angle and an exterior angle of the indicated
polygon.
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9.
Regular Octagon
10.
Regular 100-gon
11.
The side view of a storage shed is shown below. Find the value of . Then
determine the measure of each angle.
8.2 NOTES
Properties of Parallelograms
DEFINITION: A parallelogram is a quadrilateral with __________________________________
_____________________________________________________________________________.
You can call this figure: “Parallelogram PQRS” or.
In
,
and
, by definition.
Other properties: If a quadrilateral is a parallelogram, then….
THM 8.3
THM 8.4
THM 8.5
THM 8.6
Ex1: Find the values of x and y.
Your thoughts:
1) Is it a parallelogram? YES, b/c the opp. sides
are parallel (def.)
2) Opposite sides are congruent, so x + 4 = 12.
3) Opposite angles are congruent, so y = 65.
Ex 2:
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Ex 3-6:
Ex 7: The measure of one interior angle of a parallelogram is 50 degrees more than 4
times the measure of another angle. Find the measure of each angle.
(Make a sketch and label it.)
Ex 8: In LMNO , the ratio of LM to MN is 4:3. Find LM if the perimeter of LMNO is 28.
Ex 9: The diagonals of LMNO intersect at point P. What are the coordinates of P?
Hint: What do you need to know?
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Ex 10: Is the quadrilateral formed by the lines on the graph a parallelogram?
Hint: What do you need to know?
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8.2 HOMEWORK
Find the measure of the indicated angle in the parallelogram.
1.
Find
2. Find
Find the value of each variable in parallelogram.
3.
4.
Find the indicated measure in
5.
6.
7.
8.
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Use the diagram of
Points
Find the indicated measure.
and
are midpoints of
and
9.
10.
11.
12.
13.
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In parallelogram
is .
the ratio of
to
is
. Find
if the perimeter of
8.3 NOTES
Proving a Quadrilateral is a Parallelogram
Examples:
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8.3 HOMEWORK
What theorem can you use to show that the quadrilateral is a parallelogram?
1.
2.
3.
4.
For what value of
is the quadrilateral a parallelogram?
5.
6.
7.
8.
9.
10.
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The vertices of quadrilateral
are given. Draw
and show that it is a parallelogram.
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11.
(
)
(
12.
(
)
(
)
(
)
)
(
(
)
)
(
)
in a coordinate plane
8.4NOTES
Rhombuses, Squares and Rectangles
In this lesson, you will learn about three special types of parallelograms:
Rhombus
A parallelogram with four
congruent sides
(equilateral).
Rectangle
Square
A parallelogram with four A parallelogram with four
right angles (equiangular). congruent sides and four
right angles (regular).
SPECIAL NOTES ABOUT SQUARES:
 Since a square has four congruent sides, it is also a _____________________________________ .
 Since a square has four right angles, it is also a __________________________________________ .
Diagonals of Rhombuses and Rectangles
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Examples:
Name each quadrilateral—parallelogram, rectangle, rhombus, and square—for
which the statement is true.
1. It is equiangular.
2. It is equiangular and equilateral.
3. It is diagonals are perpendicular.
4. Opposite sides are congruent.
5. The diagonals bisect each other.
6. The diagonals bisect opposite angles.
Classify the special quadrilateral. Explain your reasoning.
Then find the values of and .
7.
9.
8.
In the rhombus to the right, given
__________
__________
__________
__________
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. Find all the other angles.
10. Given rectangle CERT and
, find each measure.
__________
__________
__________
__________
11. Given rectangle CERT,
(1)
If
(2)
If
and
, solve for .
and
, solve for .
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8.4 HOMEWORK
For any rhombus
decide whether the statement is always or sometimes true.
Draw a diagram and explain your reasoning.
1.
2.
For any rectangle
decide whether the statement is always or sometimes true.
Draw a diagram and explain your reasoning.
3.
4.
Classify the quadrilateral. Explain your reasoning.
5.
6.
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Classify the special quadrilateral. Explain your reasoning. Then find the values of
and
7.
8.
The diagonals of rhombus
intersect at
find the indicated measure.
Given that
and
9.
10.
11.
12.
13.
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In preparation for a storm, a window is protected by nailing boards along its
diagonals. The lengths of the boards are the same. Can you conclude that the
window is a square? Explain.
8.5/8.6 NOTES
Trapezoids & Kites
DEFINITION: A trapezoid is a quadrilateral with ________________________________________
_________________________________________________________________________________.
 The parallel sides are called the __________________________________.
 The non-parallel sides are called the __________________________________.
 Since a trapezoid has two bases, it has two pairs of __________________________________.
DEFINITION: An isosceles trapezoid is one in which the __________________________________.
 Think of it as an isosceles triangle with the “top” cut off by a segment parallel to a
base.
*An isosceles trapezoid has:
(THM 8.14)  ___________________________________________________
(THM 8.16)
 ___________________________________________________
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DEFINITION: The midsegment of a trapezoid is a segment that connects the
____________________________ of its ____________________________.
(THM 8.17)
The midsegment of a trapezoid is parallel to each base and measures on
half the sum of the base lengths.
If MN is the midsegment of trapezoid 𝐴𝐵𝐶𝐷,
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then MN AB, MN DC, and 𝑀𝑁 2 (𝐴𝐵 𝐶𝐷)
**In other words,
Examples:
1. Find
3. Solve for .
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and
.
2. Find the length of the midsegment.
4. In trapezoid
,
and
.
The midsegment is
.
Sketch
and its midsegment and find
.
DEFINITION: A kite is a quadrilateral with ____________________________________________________.
THREE RULES FOR KITES
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UNIT 8 PERFORMANCE TASK
Performance of a Lifetime
You are a dance choreographer and have been asked by the Houston Rockets to come up
with a 45 second dance that will be showcased during a playoff game. You have been
allowed to use the entire court for the performance and have been asked to make sure
that all sides of the viewing audience will be able to see the performance without using the
Jumbotron. Your goal is to fill the 94 x 50 foot court by placing the dancers in the shape of
a polygon. You want to make sure that you use as much of the court as possible while
making sure that there is an equal amount of unused space at each corner of the court.
1. Determine where the vertices of your polygon must fall to develop an equal amount of
unused space so all fans can see the performance and the owner of the Houston Rockets
will be happy. How would you find these points and how do you know the unused space
is equal at each corner of the court?
2. What type of polygon is created? How can you justify your answer? Justify all of the
polygon’s properties.
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