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C H A P T E R
Quadrilaterals
© 2010 Carnegie Learning, Inc.
8
8
Carpenters build, install, and maintain wooden objects such as buildings and
furniture. There are almost 1.5 million carpenters in the United States, making it
the nation's largest building trades occupation. When building, carpenters often
use math, along with specialized tools such as levels and squares. You will use
math to calculate whether a bookcase was properly built.
8.1
Squares and Rectangles
8.4
Properties of Squares and
Rectangles | p. 417
8.2
Parallelograms and Rhombi
Sum of the Interior Angle Measures
of a Polygon | p. 451
8.5
Properties of Parallelograms and
Rhombi | p. 429
8.3
Exterior and Interior Angle
Measurement Interactions
Sum of the Exterior Angle Measures
of a Polygon | p. 459
Kites and Trapezoids
Properties of Kites and
Trapezoids | p. 437
Decomposing Polygons
8.6
Quadrilateral Family
Categorizing Quadrilaterals | p. 467
Chapter 8
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Quadrilaterals
413
Introductory Problem for Chapter 8
Ms. Smith was teaching her class how to make Venn diagrams. She used the
following example.
S
2
7
L
FH
6
8
5
4
3
T
1
Rectangle S represents all students in the high school.
Circle L represents all students on the lacrosse team.
Circle FH represents all students on the field hockey team.
Circle T represents all students on the track team.
The numbers on the Venn diagram represent 8 different regions.
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Region 2:
Region 3:
Region 4:
Region 5:
Region 6:
Region 7:
Region 8:
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© 2010 Carnegie Learning, Inc.
1. Describe which sport(s) the students in each designated region play.
Region 1:
2. Create a Venn diagram that describes the relationship between all of the terms
listed. Number each region and name the polygons found in each region. Use
RH for rhombus and RE for rectangle.
Trapezoids
Kites
Rhombi
Rectangles
Parallelograms
Squares
Quadrilaterals
© 2010 Carnegie Learning, Inc.
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3. Write a description for each numbered region.
Be prepared to share your solutions and methods.
Chapter 8
|
Introductory Problem for Chapter 8
415
© 2010 Carnegie Learning, Inc.
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416
Chapter 8
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Quadrilaterals
8.1
Squares and Rectangles
Properties of Squares and Rectangles
OBJECTIVES
KEY TERM
In this lesson you will:
l
l
l
l
l
Prove the Perpendicular/Parallel Line Theorem.
Construct a square and a rectangle.
Determine the properties of a square and rectangle.
Prove the properties of a square and a rectangle.
PROBLEM 1
Perpendicular/Parallel
Line Theorem
Know the Properties or be Square!
A quadrilateral is a four-sided polygon. A diagonal of a polygon is a line segment
connecting two non-adjacent vertices.
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A square is a quadrilateral with four right angles and all sides congruent.
© 2010 Carnegie Learning, Inc.
Quadrilaterals have different properties directly related to the measures of their
interior angles and lengths of their sides. Perpendicular lines and right angles are
useful when proving properties of certain quadrilaterals.
1. Devon is trying to think of quadrilaterals that have four right angles.
Can you help him?
Lesson 8.1
|
Squares and Rectangles
417
2. Ramira is helping Jessica with her math homework. She tries to explain the
theorem: “If two lines are perpendicular to the same line, then the two lines are
parallel to each other.” Jessica doesn’t understand why this is true. Use the
diagram shown and complete the proof to help Jessica understand this theorem.
ᐉ3
1 2
ᐉ2
3 4
ᐉ1
5 6
7 8
Given: ᐉ1 ⊥ ᐉ3; ᐉ2 ⊥ ᐉ3
Prove: ᐉ1 ᐉ2
Statements
Reasons
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© 2010 Carnegie Learning, Inc.
Perpendicular/Parallel Line Theorem: If two lines are perpendicular to the same
line, then the two lines are parallel to each other.
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3. Draw a square with two diagonals. Label the vertices and the intersection of the
diagonals. List all of the properties you know to be true.
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___
___
___
4. Use AB to construct square ABCD with diagonals AC and BD intersecting at
point E.
B
© 2010 Carnegie Learning, Inc.
A
Lesson 8.1
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Squares and Rectangles
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5. Create a two-column proof of the statement DAB CBA.
D
C
E
A
___
___
B
Given: Square ABCD with diagonals AC and BD intersecting at point E
Prove: DAB CBA
Statements
8
Reasons
___ ___
Congratulations! You have just proven a property of a square.
Property of a Square: Diagonals of a square are congruent.
You can now use this property as a valid reason in future proofs.
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6. Do you have enough information to conclude AC BD ? Explain.
___ ___
___ ___
7. Create a two-column proof of the statement DA CB and DC AB .
D
C
A
B
Given: Square
___ ___ABCD___ ___
Prove: DA CB and DC AB
Statements
Reasons
8
© 2010 Carnegie Learning, Inc.
8. If a parallelogram is a quadrilateral with opposite sides parallel, do you have
enough information to conclude square ABCD is a parallelogram? Explain.
Congratulations! You have just proven another property of a square!
Property of a Square: Opposite sides of a square are parallel.
You can now use this property as a valid reason in future proofs.
Lesson 8.1
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Squares and Rectangles
421
___ ___
___ ___
9. Create a two-column proof. Use DEC and BEA to prove DE BE and CE AE .
D
C
E
A
___
___
B
Given: Square
with diagonals
AC and BD intersecting at point E
___ ___ABCD ___
___
Prove: DE BE and CE AE
Statements
Reasons
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Congratulations! You have just proven another property of a square!
Property of a Square: The diagonals of a square bisect each other.
You can now use this property as a valid reason in future proofs.
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10. Do you have enough information to conclude the diagonals of a square bisect
each other? Explain.
11. Write a paragraph proof to conclude the diagonals of a square bisect the vertex
angles. Use the square in Question 9 and the Property of a Square:
The diagonals of a square bisect each other.
Congratulations! You have just proven another property of a square!
Property of a Square: Opposite sides of a square are parallel.
You can now use this property as a valid reason in future proofs.
12. Write a paragraph proof to conclude the diagonals of a square are
perpendicular to each other. Use the square in Question 9.
© 2010 Carnegie Learning, Inc.
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Congratulations! You have just proven another property of a square!
Property of a Square: The diagonals of a square are perpendicular to each other.
You can now use this property as a valid reason in future proofs.
13. Revisit Question 3 to make sure you have listed all of the properties of
a square.
Lesson 8.1
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Squares and Rectangles
423
PROBLEM 2
The Rectangle
A rectangle is a quadrilateral with opposite sides congruent and all
angles congruent.
1. Draw a rectangle with two diagonals. Label the vertices and the intersection of
the two diagonals. List all of the properties you know to be true. (Do not draw
a square.)
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___
___
R
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E
© 2010 Carnegie Learning, Inc.
___
2. Use RE to construct rectangle RECT with diagonals RC and ET intersecting at
point A. Do not construct a square.
3. Create a two-column proof of the statement RCT ETC.
R
E
A
T
___
C
___
Given: Rectangle RECT with diagonals RC and ET intersecting at point A
Prove: RCT ETC
Statements
Reasons
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© 2010 Carnegie Learning, Inc.
___ ___
4. Do you have enough information to conclude RT EC ? Explain.
Lesson 8.1
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Squares and Rectangles
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5. In a paragraph proof, describe how you could prove the second pair of
opposite sides of the rectangle are congruent.
6. Do you have enough information to conclude rectangle RECT is a
parallelogram? Explain your reasoning.
7. Do you have enough information to conclude the diagonals of a rectangle
are congruent?
8. Do you have enough information to conclude the diagonals of a rectangle
bisect each other? Explain.
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© 2010 Carnegie Learning, Inc.
9. Revisit Question 1 of Problem 2 and make sure you have listed all of the
properties of a rectangle.
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PROBLEM 3
Application Problems
1. Ofelia is making a square mat for a picture frame. How can she make sure the
mat is a square using only a ruler?
2. Gretchen is putting together a bookcase. It came with diagonal support bars
that are to be screwed into the top and bottom on the back of the bookcase.
Unfortunately, the instructions were lost and Gretchen does not have the
directions or a measuring tool. She has a screwdriver, a marker, and a piece
of string. How can Gretchen attach the supports to make certain the bookcase
will be a rectangle and the shelves are parallel to the ground?
© 2010 Carnegie Learning, Inc.
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Lesson 8.1
|
Squares and Rectangles
427
3. Matsuo knows this birdhouse has a rectangular base, but he wonders if it has a
square base.
a. What does Matsuo already know to conclude the birdhouse has a
rectangular base?
b. What does Matsuo need to know to conclude the birdhouse has a
square base?
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© 2010 Carnegie Learning, Inc.
Be prepared to share your solutions and methods.
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8.2
Parallelograms and Rhombi
Properties of Parallelograms and Rhombi
OBJECTIVES
KEY TERM
In this lesson you will:
l
Construct a parallelogram.
Construct a rhombus.
l Prove the properties of a parallelogram.
l Prove the properties of a rhombus.
l
Parallelogram/Congruent-Parallel
Side Theorem
l
PROBLEM 1
The Parallelogram
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
8
© 2010 Carnegie Learning, Inc.
1. Draw a parallelogram with two diagonals. Label the vertices and the
intersection of the diagonals. List all of the properties you know to be true.
Do not draw a square or a rectangle.
Lesson 8.2
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Parallelograms and Rhombi
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___
___
___
2. Use PA to construct parallelogram PARG with diagonals PR and AG
intersecting at point M.
A
P
3. To prove opposite sides of a parallelogram are congruent, which triangles
would you prove congruent?
P
A
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M
R
© 2010 Carnegie Learning, Inc.
G
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4. Use PGR and RAP in the parallelogram from Question 3 to prove opposite
sides of a parallelogram
Create a two-column proof of the
___ ___ are
___congruent.
___
statement PG AR and GR PA .
___ ___
Given: Parallelogram
___ ___ PARG
___ with
___ diagonals PR and AG intersecting at point M
Prove: PG AR and GR PA
Statements
Reasons
Congratulations! You have just proven a property of a parallelogram!
Property of a Parallelogram: Opposite sides of a parallelogram are congruent.
You can now use this property as a valid reason in future proofs.
5. Do you have enough information to conclude PGR RAP ? Explain.
© 2010 Carnegie Learning, Inc.
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6. What additional angles would you need to show congruent to prove opposite
angles of a parallelogram are congruent? What two triangles do you need to
prove congruent?
Lesson 8.2
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Parallelograms and Rhombi
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7. Use APG and GRA in the diagram from Question 3 to prove opposite
angles of a parallelogram are congruent. Create a two-column proof of the
statement GPA ARG.
___
___
Given: Parallelogram PARG with diagonals PR and AG intersecting at point M
Prove: GPA ARG
(You have already proven PGR RAP in Question 5.)
Statements
Reasons
Congratulations! You have just proven another property of a parallelogram!
Property of a Parallelogram: Opposite angles of a parallelogram are congruent.
You can now use this property as a valid reason in future proofs.
8
Congratulations! You have just proven another property of a parallelogram!
Property of a Parallelogram: The diagonals of a parallelogram bisect each other.
You can now use this property as a valid reason in future proofs.
9. Ray told his math teacher that he thinks a quadrilateral is a parallelogram if only
one pair of opposite sides is known to be both congruent and parallel. Is Ray
correct? Write a paragraph proof to verify Ray’s conjecture. Use the diagram
from Question 3.
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© 2010 Carnegie Learning, Inc.
8. Write a paragraph proof to conclude the diagonals of a parallelogram bisect
each other. Use the parallelogram in Question 3.
Parallelogram/Congruent-Parallel Side Theorem: If one pair of opposite sides of a
quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.
10. Revisit Question 1 to make sure you have listed all of the properties of a
parallelogram.
PROBLEM 2
The Rhombus
A rhombus is a quadrilateral with all sides congruent.
1. Draw a rhombus with two diagonals. Label the vertices and the intersection of
the two diagonals. List all of the properties you know to be true. (Do not draw
a square.)
© 2010 Carnegie Learning, Inc.
8
Lesson 8.2
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Parallelograms and Rhombi
433
___
___
___
2. Use RH to construct rhombus RHOM with diagonals RO and HM intersecting
at point B. Do not construct a square.
H
R
3. Write a paragraph proof to conclude rhombus RHOM is a parallelogram.
R
H
B
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O
Congratulations! You have just proven a property of a rhombus!
Property of a Rhombus: A rhombus is a parallelogram.
You can now use this property as a valid reason in future proofs.
4. Since a rhombus is a parallelogram, what properties hold true for all rhombi?
5. Write a paragraph proof to conclude the diagonals of a rhombus are
perpendicular. Use the rhombus in Question 3.
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M
6. Write a paragraph proof to conclude the diagonals of a rhombus bisect the
vertex angles. Use the rhombus in Question 3.
7. Revisit Question 1 and make sure you have listed all of the properties of
a rhombus.
PROBLEM 3
Application Problems
1. Jim tells you he is thinking of a quadrilateral that is either a square or a
rhombus, but not both. He wants you to guess which quadrilateral he is
thinking of and allows you to ask one question about the quadrilateral.
Which question should you ask?
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© 2010 Carnegie Learning, Inc.
2. Mrs. Baker told her geometry students to bring in a picture of a parallelogram
for extra credit. Albert brought in a picture of the flag shown. The teacher
handed Albert a ruler and told him to prove it was a parallelogram. What are
two ways Albert could prove the picture is a parallelogram?
Lesson 8.2
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Parallelograms and Rhombi
435
3. Mrs. Baker held up two different lengths of rope shown and a piece of chalk.
She asked her students if they could use this rope and chalk to construct
a rhombus on the blackboard. Rena raised her hand and said she could
construct a rhombus with the materials. Mrs. Baker handed Rena the chalk and
rope. What did Rena do?
Be prepared to share your solutions and methods.
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8.3
Kites and Trapezoids
Properties of Kites and Trapezoids
OBJECTIVES
KEY TERMS
In this lesson you will:
l
l
l
l
l
l
Construct a kite.
Construct a trapezoid.
Prove the properties of a kite.
Prove the properties of a trapezoid.
Prove a biconditional statement.
PROBLEM 1
l
l
base angles of a trapezoid
isosceles trapezoid
biconditional statement
Let's Go Fly a Kite!
A kite is a quadrilateral with two pairs of consecutive congruent sides with opposite
sides that are not congruent.
8
© 2010 Carnegie Learning, Inc.
1. Draw a kite with two diagonals. Label the vertices and the intersection of the
two diagonals. List all of the properties you know to be true.
Lesson 8.3
|
Kites and Trapezoids
437
__
___
2. Construct kite KITE with diagonals IE and KT intersecting at point S.
3. To prove one pair of opposite angles of a kite is congruent, which triangles in
the kite would you prove congruent? Explain your reasoning.
I
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K
S
T
4. Prove one pair of opposite angles
___of a kite
__ congruent.
Given: Kite KITE with diagonals KT and IE intersecting at point S.
Prove: KIT KET
Statements
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Reasons
© 2010 Carnegie Learning, Inc.
E
Congratulations! You have just proven a property of a kite!
Property of a Kite: One pair of opposite angles is congruent.
You are now able to use this property as a valid reason in future proofs.
___
5. Do you have enough information to conclude KT bisects IKE and ITE?
Explain your reasoning.
__ ___
6. What two triangles could you use to prove IS ES ?
__ ___
7. If IS ES , is that enough information to determine that one diagonal of a kite
bisects the other diagonal? Explain.
8
© 2010 Carnegie Learning, Inc.
8. Write a paragraph proof to conclude the diagonals of a kite are perpendicular
to each other.
Congratulations! You have just proven another property of a kite!
Property of a Kite: The diagonals of a kite are perpendicular to each other.
You are now able to use this property as a reason in future proofs.
9. Revisit Question 1 to make sure you have listed all of the properties of a kite.
Lesson 8.3
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Kites and Trapezoids
439
PROBLEM 2
The Trapezoid
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
The bases of a trapezoid are its parallel sides. The base angles of a trapezoid are
either pair of angles that share a base as a common side. The legs of a trapezoid are
its non-parallel sides.
1. Draw a trapezoid. Identify the vertices, bases, base angles, and legs.
___
2. Use TR to construct trapezoid TRAP.
T
R
© 2010 Carnegie Learning, Inc.
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An isosceles trapezoid is a trapezoid with congruent non-parallel sides.
One property of an isosceles trapezoid is that the base angles of an isosceles
trapezoid are congruent.
3. Create a two-column proof of this property of an isosceles trapezoid. You will
need to draw an auxiliary line parallel to one of the congruent legs to prove this
property. You will also need to do the proof in two parts because there are two
pairs of base angles.
___ ___ ___ ___
Given: Isosceles Trapezoid TRAP with TR PA , TP RA
Prove: T R
T
Z
P
Statements
R
A
Reasons
© 2010 Carnegie Learning, Inc.
8
You must also prove A TPA. Write a paragraph proof to prove
A TPA.
Lesson 8.3
|
Kites and Trapezoids
441
4. Kala insists that if a trapezoid has only one pair of congruent base angles, then
the trapezoid must be isosceles. She thinks proving two pairs of base angles
are congruent is not necessary. Prove the given statement using a two column
proof to show that Kala is correct.
___ ___
Given: Isosceles
Trapezoid
TRAP
with
TR
PA , T R
___ ___
Prove: TP RA
T
Z
P
R
A
Statements
Reasons
An if and only if statement is called a biconditional statement because it consists of
two separate conditional statements rewritten as one statement. It is a combination
of both a conditional statement and the converse of that conditional statement. A
biconditional statement is true only when the conditional statement and the converse
of the statement are both true.
Consider the following property of an isosceles trapezoid:
The diagonals of an isosceles trapezoid are congruent.
The property clearly states that if a trapezoid is isosceles, then the diagonals are
congruent. Is the converse of this statement true? If so, then this property can be
written as a biconditional statement. Rewording the property as a biconditional
statement becomes:
“A trapezoid is isosceles if and only if its diagonals are congruent.”
To prove this biconditional statement is true, rewrite it as two conditional statements
and prove each statement.
Statement 1: If a trapezoid is an isosceles trapezoid, then the diagonals of the
trapezoid are congruent. (Original statement)
Statement 2: If the diagonals of a trapezoid are congruent, then the trapezoid is an
isosceles trapezoid. (Converse of original statement)
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5. Use the trapezoid shown to prove each statement.
T
R
A
P
___ ___ ___ ___
Given: Isosceles Trapezoid
___
___TRAP with TP RA , TR PA , and
diagonals
TA
and
PR
.
___ ___
Prove: TA PR
Statements
Reasons
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___ ___
___
___
© 2010 Carnegie Learning, Inc.
Given: Trapezoid TRAP with TP RA , and diagonals TA PR
Prove: Trapezoid TRAP is isosceles
___
To prove the converse, auxiliary lines must be drawn such that RA is extended
to
___intersect
___ a perpendicular line passing through point T perpendicular to
RA (TE ) and intersect
a second perpendicular line passing through point P
___ ___
perpendicular to RA (PZ ).
T
E
R
A
P
Z
Notice that quadrilateral TEZP is a rectangle.
Lesson 8.3
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Kites and Trapezoids
443
6. Write a paragraph proof to prove the converse is true.
The property of an isosceles trapezoid can now be written as a biconditional
statement because the conditional statement and its converse have both been
proven to be true.
Property of an Isosceles Trapezoid: A trapezoid is isosceles if and only if its
diagonals are congruent.
PROBLEM 3 Construction
Segment AD is the perimeter of an isosceles trapezoid. Follow the steps to construct
the isosceles trapezoid.
8
A
___
Choose a segment on segment AD for the shorter base (___
AB ).
Choose a segment on segment AD for the longer base ( BC ).
Segment
___CD represents the sum of the length of the two legs.
Bisect CD to determine the length of each congruent leg. Label the midpoint E.
© 2010 Carnegie Learning, Inc.
1.
2.
3.
4.
D
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5. Copy segment AB onto segment
___BC (creating segment BF ) to determine the
difference between the bases (FC ).
___
6. Bisect FC to determine point G (FG ⫽ CG).
© 2010 Carnegie Learning, Inc.
8
Lesson 8.3
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Kites and Trapezoids
445
___
7. Take FG (half the
___ difference of the base
___ lengths) and copy it onto the left end of
the long base BC (creating distance
____ BH ). Notice that it is already marked off on
the right end of the long base (GC ).
8. Construct the perpendicular through point H. Note the distance between the
two most left perpendiculars is the length of the short base.
© 2010 Carnegie Learning, Inc.
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9. Place the compass point on B and mark off the distance of one leg (CE) on
the left most perpendicular, name the new point I. This forms one leg of the
isosceles trapezoid.
10. Place the compass point on C and mark off the length of one leg (CE) on the
other perpendicular, name the new point J. This is one leg of the isosceles
trapezoid. Note that IJ ⫽ AB. BIJC is an isosceles trapezoid!
© 2010 Carnegie Learning, Inc.
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Lesson 8.3
|
Kites and Trapezoids
447
PROBLEM 4
Application Problems
1. Solve for the perimeter of the kite.
© 2010 Carnegie Learning, Inc.
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Quadrilaterals
2. Could quadrilaterals 1, 2, and 3 on this kite be squares? Explain.
2
3
1
5
4
© 2010 Carnegie Learning, Inc.
8
Lesson 8.3
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Kites and Trapezoids
449
3. Trevor used a ruler to measure the height of each trapezoid and the length of
each leg. He tells Carmen the three trapezoids must be congruent because
they are all the same height and have congruent legs. What does Carmen need
to do to convince Trevor that he is incorrect?
8
© 2010 Carnegie Learning, Inc.
Be prepared to share your solutions and methods.
450
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Quadrilaterals
8.4
Decomposing Polygons
Sum of the Interior Angle Measures
of a Polygon
KEY TERM
OBJECTIVES
In this lesson you will:
l Write a formula for the sum of the interior
angles of any polygon.
l Calculate the sum of the interior angles of any
polygon, given the number of sides.
l Calculate the number of sides of a polygon,
given the sum of the interior angles.
l Write a formula for the measure of each interior
angle of any regular polygon.
l Calculate the measure of an interior angle of a
regular polygon, given the number of sides.
l Calculate the number of sides of a regular
polygon, given the sum of the interior angles.
© 2010 Carnegie Learning, Inc.
PROBLEM 1
l
interior angle of a polygon
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Who’s Correct?
In geometry, it is necessary to know the sum of the interior angles of various
polygons to determine other information. An interior angle of a polygon is an
angle which faces the inside of a polygon and is formed by consecutive sides of the
polygon. Is there a quick method for calculating the sum of the measures of different
polygons? Let’s find out!
Ms. Lambert asked her class to determine the sum of the interior angle measures
of a quadrilateral. She that two of her students, Carson and Juno, were already
engaged in a heated disagreement.
Lesson 8.4
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Decomposing Polygons
451
Carson drew a quadrilateral and added one diagonal as shown. He concluded
that the sum of the measures of the interior angles of a quadrilateral must be
equal to 360º.
1. Describe Carson’s reasoning.
Juno drew a quadrilateral and added two diagonals as shown. She concluded
that the sum of the measures of the interior angles of a quadrilateral must be equal
to 720º.
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3. Who is correct? Explain.
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2. Describe Juno’s reasoning.
PROBLEM 2
The Sum of the Interior Angle Measures
of a Polygon
As always, you must start with what you know to be true. The Triangle Sum
Theorem states that the sum of the three interior angles of any triangle is equal to
180º. You can use this information to calculate the sum of the interior angles of other
polygons.
1. Calculate the sum of the interior angle measures of a quadrilateral by
completing each step.
Step 1: Draw a quadrilateral.
Step 2: Draw all possible diagonals using only one vertex of the quadrilateral.
Remember, a diagonal is a line segment connecting non-adjacent
vertices.
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© 2010 Carnegie Learning, Inc.
Step 3: How many triangles are formed when the diagonal(s) divide the
quadrilateral?
Step 4: If the sum of the interior angle measures of each triangle is 180º, what
is the sum of all the interior angle measures of the triangles formed by
the diagonal(s)?
Lesson 8.4
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Decomposing Polygons
453
2. Calculate the sum of the interior angle measures of a pentagon by completing
each step.
Step 1: Draw a pentagon.
Step 2: Draw all possible diagonals using only one vertex of the pentagon.
Step 3: How many triangles are formed when the diagonal(s) divide the
pentagon?
8
3. Calculate the sum of the interior angle measures of a hexagon by completing
each step.
Step 1: Draw a hexagon.
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Step 4: If the sum of the interior angle measures of each triangle is 180º, what
is the sum of all the interior angle measures of the triangles formed by
the diagonal(s)?
Step 2: Draw all possible diagonals using one vertex of the hexagon.
Step 3: How many triangles are formed when the diagonal(s) divide
the hexagon?
Step 4: If the sum of the interior angle measures of each triangle is 180º, what
is the sum of all the interior angle measures of the triangles formed by
the diagonal(s)?
4. Complete the table shown.
Number of sides of the polygon
3
4
5
6
Number of diagonals drawn
8
Number of triangles formed
© 2010 Carnegie Learning, Inc.
Sum of the measures of the interior angles
5. What pattern do you notice between the number of possible diagonals drawn
from one vertex of the polygon, and the number of triangles formed by those
diagonals?
6. Compare the number of sides of the polygon to the number of possible
diagonals drawn from one vertex. What do you notice?
7. Compare the number of sides of the polygon to the number of triangles formed
by drawing all possible diagonals from one vertex. What do you notice?
Lesson 8.4
|
Decomposing Polygons
455
8. What pattern do you notice about the sum of the interior angle measures of a
polygon as the number of sides of each polygon increases by 1?
9. Predict the number of possible diagonals drawn from one vertex and the
number of triangles formed for a seven-sided polygon using the table you
completed.
10. Predict the sum of all the interior angle measures of a seven-sided polygon
using the table your completed.
11. Continue the pattern to complete the table.
Number of sides of the polygon
7
8
9
16
Number of diagonals drawn
Number of triangles formed
Sum of the measures of the interior angles
12. When you calculated the number of triangles formed in the 16-sided polygon,
did you need to know how many triangles were formed in a 15-sided polygon
first? Explain your reasoning.
8
14. What is the sum of all the interior angle measures of a 100-sided polygon?
Explain your reasoning.
15. If a polygon has n sides, how many triangles are formed by drawing all
diagonals from one vertex? Explain.
16. What is the sum of all the interior angle measures of an n-sided polygon?
Explain your reasoning.
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13. If a polygon has 100 sides, how many triangles are formed by drawing all
possible diagonals from one vertex? Explain.
17. Use the formula to calculate the sum of all the interior angle measures of a
polygon with 32 sides.
18. If the sum of all the interior angle measures of a polygon is 9540º,
how many sides does the polygon have? Explain your reasoning.
PROBLEM 3
Sum of the Interior Angle Measures
of a Regular Polygon
1. Use the formula developed in Problem 2, Question 16 to calculate the sum of
the all the interior angle measures of a decagon.
2. Calculate each interior angle measure of a decagon if each interior angle is
congruent. How did you calculate your answer?
8
3. Complete the table.
Number of sides of regular polygon
3
4
5
6
7
8
© 2010 Carnegie Learning, Inc.
Sum of measures of interior angles
Measure of each interior angle
4. If a regular polygon has n sides, write a formula to calculate the measure of
each interior angle.
5. Use the formula to calculate each interior angle measure of a regular
100-sided polygon.
Lesson 8.4
|
Decomposing Polygons
457
6. If each interior angle measure of a regular polygon is equal to 150º, determine
the number of sides. How did you calculate your answer?
7. Apply what you have learned about the interior angle measures of a regular
polygon on the star shown. PENTA is a regular pentagon. Solve for x.
P
E
x
A
N
T
Be prepared to share your methods and solutions.
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8
8.5
Exterior and Interior Angle
Measurement Interactions
Sum of the Exterior Angle Measures of a Polygon
OBJECTIVES
KEY TERM
In this lesson you will:
l
exterior angle of a polygon
Write a formula for the sum of the exterior angles
of any polygon.
l Calculate the sum of the exterior angles of any
polygon, given the number of sides.
l Write a formula for the measure of each exterior
angle of any regular polygon.
l Calculate the measure of an exterior angle of a
regular polygon, given the number of sides.
l Calculate the number of sides of a regular polygon,
given the measure of each exterior angle.
l
8
© 2010 Carnegie Learning, Inc.
PROBLEM 1
Is There a Formula?
In the previous lesson, you wrote a formula for the sum of all the interior angle
measures of a polygon. In this lesson, you will write a formula for the sum of all the
exterior angle measures of a polygon.
Lesson 8.5
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Exterior and Interior Angle Measurement Interactions
459
Each interior angle of a polygon can be paired with an exterior angle. An exterior
angle of a polygon is formed adjacent to each interior angle by extending one side
of each vertex of the polygon as shown in the triangle. Each exterior angle and the
adjacent interior angle form a linear pair.
Exterior
Angle
Exterior
Angle
Exterior
Angle
1. Use the formula for the sum of interior angle measures of a polygon and the
Linear Pair Postulate to calculate the sum of the exterior angle measures of
a triangle.
© 2010 Carnegie Learning, Inc.
8
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2. Calculate the sum of the exterior angle measures of a quadrilateral by
completing each step.
Step 1: Draw a quadrilateral and extend each side to locate an exterior angle
at each vertex.
Step 2: Use the formula for the sum of interior angle measures of a polygon
and the Linear Pair Postulate to calculate the sum of the exterior angle
measures of a quadrilateral.
© 2010 Carnegie Learning, Inc.
8
3. Calculate the sum of the exterior angle measures of a pentagon by completing
each step.
Step 1: Draw a pentagon and extend each side to locate an exterior angle at
each vertex.
Lesson 8.5
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Exterior and Interior Angle Measurement Interactions
461
Step 2: Use the formula for the sum of the interior angle measures of a
polygon and the Linear Pair Postulate to calculate the sum of the
exterior angle measures of a pentagon.
4. Calculate the sum of the exterior angle measure of a hexagon by completing
each step.
Step 1: Without drawing a hexagon, how many linear pairs are formed by
each interior and adjacent exterior angle? How do you know?
8
Step 3: Use the formula for the sum of the interior angle measures of a
polygon and the Linear Pair Postulate to calculate the sum of the
measures of the exterior angles of a hexagon.
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Step 2: What is the relationship between the number of sides of a polygon
and the number of linear pairs formed by each interior angle and its
adjacent exterior angle?
5. Complete the table.
Number of sides of
the polygon
3
4
5
6
7
15
Number of linear pairs
formed
Sum of measures of
linear pairs
Sum of measures of
interior angles
Sum of measures of
exterior angles
6. When you calculated the sum of the exterior angle measures in the 15-sided
polygon, did you need to know anything about the number of linear pairs, the
sum of the linear pair measures, or the sum of the interior angle measures of
the 15-sided polygon? Explain.
© 2010 Carnegie Learning, Inc.
7. If a polygon has 100 sides, calculate the sum of the exterior angle measures.
Explain how you calculated your answer.
8
8. What is the sum of the exterior angle measures of an n-sided polygon?
9. If the sum of the exterior angle measures of a polygon is 360⬚, how many sides
does the polygon have? Explain how you got this answer.
Lesson 8.5
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Exterior and Interior Angle Measurement Interactions
463
10. Explain why the sum of the exterior angle measures of any polygon is always
equal to 360⬚.
PROBLEM 2
Regular Polygons
1. Calculate the measure of each exterior angle of an equilateral triangle.
Explain your reasoning.
2. Calculate the measure of each exterior angle of a square. Explain your
reasoning.
8
4. Calculate the measure of each exterior angle of a regular hexagon.
Explain your reasoning.
5. Complete the table shown to look for a pattern.
Number of sides of
a regular polygon
Sum of measures
of exterior angles
Measure of each
interior angle
Measure of each
exterior angle
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3
4
5
6
7
15
© 2010 Carnegie Learning, Inc.
3. Calculate the measure of each exterior angle of a regular pentagon.
Explain your reasoning.
6. When you calculated the measure of each exterior angle in the 15-sided regular
polygon, did you need to know anything about the measure of each interior
angle? Explain.
7. If a regular polygon has 100 sides, calculate the measure of each exterior
angle. Explain how you calculated your answer.
8. What is the measure of each exterior angle of an n-sided regular polygon?
9. If the measure of each exterior angle of a regular polygon is 18⬚, how many
sides does the polygon have? Explain how you calculated your answer.
© 2010 Carnegie Learning, Inc.
8
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Exterior and Interior Angle Measurement Interactions
465
PROBLEM 3
Old Sibling Rivalry
Two sisters, Molly and Lily, were arguing about who was better at using a compass
and a straightedge.
1. Molly challenged Lily to construct a regular hexagon. Undaunted by the
challenge, Lily took the compass and went to work. What did Lily do?
2. Lily then challenged Molly to construct a square. Molly grabbed her compass
with gusto and began the construction. What did Molly do?
3. Both sisters were now glaring at each other and their mother, a math teacher,
walked into the room. Determined to end this dispute, she gave her daughters
a challenge. She told them the only way to settle the argument was to see who
could be the first to come up with a construction for a regular pentagon. Give it
a try! Hint: There are only six steps. The first two steps are to draw a starter line
and to construct a perpendicular line.
© 2010 Carnegie Learning, Inc.
8
Be prepared to share your solutions and methods.
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8.6
Quadrilateral Family
Categorizing Quadrilaterals
OBJECTIVES
In this lesson you will:
List the properties of quadrilaterals.
Categorize quadrilaterals based upon their properties.
l Construct quadrilaterals given a diagonal.
l
l
PROBLEM 1
Characteristics of Quadrilaterals
Complete the table by placing a checkmark in the appropriate row and column to
associate each figure with its properties.
© 2010 Carnegie Learning, Inc.
Square
Rectangle
Rhombus
Kite
Parallelogram
Trapezoid
Quadrilateral
8
No parallel sides
Exactly one pair of parallel sides
Two pairs of parallel sides
One pair of sides are both congruent and parallel
Two pairs of opposite sides are congruent
Exactly one pair of opposite angles are congruent
Two pairs of opposite angles are congruent
Consecutive angles are supplementary
Diagonals bisect each other
All sides are congruent
Diagonals are perpendicular to each other
Diagonals bisect the vertex angles
All angles are congruent
Diagonals are congruent
Lesson 8.6
|
Quadrilateral Family
467
PROBLEM 2
The introduction problem to this chapter involved creating a Venn diagram
associating various quadrilaterals. The properties of each quadrilateral dictate how to
correctly position and relate each quadrilateral in a Venn diagram. To conclude this
unit, we will revisit the Venn diagram problem.
1. Create a Venn diagram describing the relationships between all of the
quadrilaterals listed. Number each region and name the figure located in each
region. Use RH for rhombus and RE for rectangle.
2. Write a description for each region.
Trapezoids
Kites
Rhombi
Rectangles
Parallelograms
Squares
Quadrilaterals
© 2010 Carnegie Learning, Inc.
8
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PROBLEM 3
True or False
Determine whether each statement is true or false. If it is false, explain why.
1. A square is also a rectangle.
2. A rectangle is also a square.
3. The base angles of a trapezoid are congruent.
4. A parallelogram is also a trapezoid.
5. A square is a rectangle with all sides congruent.
8
© 2010 Carnegie Learning, Inc.
6. The diagonals of a trapezoid are congruent.
7. A kite is also a parallelogram.
8. The diagonals of a rhombus bisect each other.
Lesson 8.6
|
Quadrilateral Family
469
PROBLEM 4
Can You Read Joe’s Mind?
Joe is thinking of a specific polygon. He has listed six hints. As you read each hint,
use deductive reasoning to try and guess Joe’s polygon. By the last hint you should
be able to read Joe’s mind.
1. The polygon has four sides.
2. The polygon has at least one pair of parallel sides.
3. The diagonals of the polygon bisect each other.
4. The polygon has opposite sides congruent.
8
6. The polygon does not have four congruent angles.
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5. The diagonals of the polygon are perpendicular to each other.
PROBLEM 5
Using Diagonals
Knowing certain properties of each quadrilateral makes it possible to construct the
quadrilateral given only a single diagonal.
____
1. Describe how you could construct parallelogram WXYZ given only diagonal WY .
___
2. Describe how you could construct rhombus RHOM given only diagonal RO .
___
3. Describe how you could construct kite KITE given only diagonal KT .
8
© 2010 Carnegie Learning, Inc.
Be prepared to share your solutions and methods.
Lesson 8.6
|
Quadrilateral Family
471
Chapter 8 Checklist
KEY TERMS
l
l
base angles of a
trapezoid (8.3)
isosceles trapezoid (8.3)
l
l
biconditional statement (8.3)
interior angle of a
polygon (8.4)
l
exterior angle of a
polygon (8.5)
l
trapezoid (8.3)
isosceles trapezoid (8.3)
THEOREMS
l
Perpendicular/Parallel Line
Theorem (8.1)
l
Parallelogram/CongruentParallel Side Theorem (8.2)
l
rhombus (8.2)
kite (8.3)
CONSTRUCTIONS
l
l
l
8
square (8.1)
rectangle (8.1)
parallelogram (8.2)
8.1
l
l
Using the Perpendicular/Parallel Line Theorem
The Perpendicular/Parallel Line Theorem states: “If two lines are perpendicular to the
same line, then the two lines are parallel to each other.”
p
m
n
Because line p is perpendicular to line m and line p is perpendicular to line n, lines m and n
are parallel.
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Example:
8.1
Determining Properties of Squares
A square is a quadrilateral with four right angles and all sides congruent. You can
use the Perpendicular/Parallel Line Theorem and congruent triangles to determine
the following properties of squares.
• The diagonals of a square are congruent.
• Opposite sides of a square are parallel.
• The diagonals of a square bisect each other.
• The diagonals of a square bisect the vertex angles.
• The diagonals of a square are perpendicular to each other.
Example:
For square PQRS, the following statements are true:
___
___
• PR QS
___
___
Q
___
R
___
• PQ || RS and PS || QR
___
___
___
___
• PT RT and QT ST
• PQS RQS, QRP SRP, RSQ PSQ,
and SPR QPR
___
T
P
S
___
• PR QS
8.1
8
Determining Properties of Rectangles
© 2010 Carnegie Learning, Inc.
A rectangle is a quadrilateral with opposite sides congruent and with four right
angles. You can use the Perpendicular/Parallel Line Theorem and congruent triangles
to determine the following properties of rectangles.
• Opposite sides of a rectangle are congruent.
• Opposite sides of a rectangle are parallel.
• The diagonals of a rectangle are congruent.
• The diagonals of a rectangle bisect each other.
Example:
For rectangle FGHJ, the following statements are true.
___
___
___
___
• FG HJ and FJ GH
___
___
___
G
H
___
• FG || HJ and FJ || GH
___
___
___
___
• FH GJ
___
___
K
• GK JK and FK HK
F
Chapter 8
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|
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8.2
Determining Properties of Parallelograms
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. You can
use congruent triangles to determine the following properties of parallelograms.
• Opposite sides of a parallelogram are congruent.
• Opposite angles of a parallelogram are congruent.
• The diagonals of a parallelogram bisect each other.
Example:
For parallelogram WXYZ, the following statements are true.
____
___
____
___
• WX YZ and WZ XY
Y
X
V
• WXY WZY and XYZ XWZ
____
___
___
___
• WV YV and XV ZV
8.2
W
Z
Using the Parallelogram/Congruent-Parallel Side Theorem
The Parallelogram/Congruent-Parallel Side Theorem states: “If one pair of opposite
sides of a quadrilateral is both congruent and parallel, then the quadrilateral is
a parallelogram.”
8
Example:
___
___
___
C
___
In quadrilateral ABCD, AB CD and AB || CD. So,
quadrilateral ABCD is a parallelogram, and thus has all
of the properties of a parallelogram.
5 cm
B
D
A
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5 cm
8.2
Determining Properties of Rhombi
A rhombus is a quadrilateral with all sides congruent. You can use congruent
triangles to determine the following properties of rhombi.
• Opposite angles of a rhombus are congruent.
• Opposite sides of a rhombus are parallel.
• The diagonals of a rhombus are perpendicular to each other.
• The diagonals of a rhombus bisect each other.
• The diagonals of a rhombus bisect the vertex angles.
Example:
For rhombus ABCD, the following statements are true:
• ABC CDA and BCD DAB
___
___
___
B
C
___
• AB || CD and BC || DA
___
___
___
___
• AC BD
___
X
___
• AX CX and BX DX
• BAC DAC, ABD CBD, BCA DCA,
and CDB ADB
8.3
A
D
8
Determining Properties of Kites
© 2010 Carnegie Learning, Inc.
A kite is a quadrilateral with two pairs of consecutive congruent sides with opposite
sides that are not congruent. You can use congruent triangles to determine the
following properties of kites.
• One pair of opposite angles of a kite is congruent.
• The diagonals of a kite are perpendicular to each other.
• The diagonal that connects the opposite vertex angles that are not congruent
bisects the diagonal that connects the opposite vertex angles that are congruent.
• The diagonal that connects the opposite vertex angles that are not congruent
bisects the vertex angles.
Example:
For kite KLMN, the following statements are true:
M
• LMN LKN
____
___
___
____
• KM LN
L
N
P
• KP MP
• KLN MLN and KNL MNL
K
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8.3
Determining Properties of Trapezoids
A trapezoid is a quadrilateral with exactly one pair of parallel sides. An isosceles
trapezoid is a trapezoid with congruent non-parallel sides. You can use congruent
triangles to determine the following properties of isosceles trapezoids:
• The base angles of a trapezoid are congruent.
• The diagonals of a trapezoid are congruent.
Example:
For isosceles trapezoid PQRS, the following statements are true:
• QPS RSP
___
Q
R
___
• PR QS
T
P
8.4
S
Determining the Sum of the Interior Angle Measures of Polygons
You can calculate the sum of the interior angle measures of a polygon by using
the formula 180°(n ⫺ 2), where n is the number of sides of the polygon. You can
calculate the measure of each interior angle of a regular polygon by dividing the
formula by n, the number of sides of the regular polygon.
8
Examples:
The sum of the interior angle measures of a pentagon is 180°(5 ⫺ 2) ⫽ 540°.
720° ⫽ 120°.
Each interior angle of a regular hexagon measures _____
6
The sum of the interior angle measures of a decagon is 180°(10 ⫺ 2) ⫽ 1440°.
1440° ⫽ 144°.
Each interior angle of a regular decagon measures ______
10
The sum of the interior angle measures of an 18-gon is 180°(18 ⫺ 2) ⫽ 2880°.
2880° ⫽ 160°.
Each interior angle of a regular 18-gon measures ______
18
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540° ⫽ 108°.
Each interior angle of a regular pentagon measures _____
5
The sum of the interior angle measures of a hexagon is 180°(6 ⫺ 2) ⫽ 720°.
8.5
Determining the Sum of the Exterior Angle Measures of Polygons
You can use the formula for the sum of the interior angle measures of a polygon and
the Linear Pair Postulate to determine that the sum of the exterior angle measures of
any polygon is 360°.
Examples:
You can use the formula for the sum of the interior angle measures of a polygon to
determine that the interior angle measures of the hexagon is 720°. Then, you can use
the Linear Pair Postulate to determine that the sum of the angle measures formed
by six linear pairs is 6(180°) ⫽ 1080°. Next, subtract the sum of the interior angle
measures from the sum of the linear pair measures to get the sum of the exterior
angle measures: 1080° ⫺ 720° ⫽ 360°.
8
© 2010 Carnegie Learning, Inc.
You can use the formula for the sum of the interior angle measures of a polygon
to determine that the interior angle measures of the nonagon is 1260°. Then, you
can use the Linear Pair Postulate to determine that the sum of the angle measures
formed by nine linear pairs is 9(180°) ⫽ 1620°. Next, subtract the sum of the interior
angle measures from the sum of the linear pair measures to get the sum of the
exterior angle measures: 1620° ⫺ 1260° ⫽ 360°.
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8.6
Identifying Characteristics of Quadrilaterals
The table shows the characteristics of special types of quadrilaterals.
Rhombus
Rectangle
Square
•
•
•
•
One pair of sides are both congruent and parallel
•
•
•
•
Two pairs of opposite sides are congruent
•
•
•
•
Exactly one pair of parallel sides
Kite
Parallelogram
Trapezoid
Two pairs of parallel sides
No parallel sides
•
•
Exactly one pair of opposite angles are congruent
•
Two pairs of opposite angles are congruent
•
•
•
•
Consecutive angles are supplementary
•
•
•
•
Diagonals bisect each other
•
•
•
•
All sides are congruent
8
Diagonals are perpendicular to each other
Diagonals bisect the vertex angles
•
•
•
•
•
•
•
•
•
Diagonals are congruent
•
•
© 2010 Carnegie Learning, Inc.
All angles are congruent
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