Download LIFEPAC® 8th Grade Math Unit 6 Worktext - HomeSchool

MATH
STUDENT BOOK
8th Grade | Unit 6
Unit 6 | Measurement
Math 806
Measurement
Introduction |3
1. Angle Measures and Circles
5
Classify and Measure Angles |5
Perpendicular and Parallel Lines, Part 1 |12
Perpendicular and Parallel Lines, Part 2 |20
Circles |25
SELF TEST 1: Angle Measures and Circles |33
2.Polygons
37
Classifying Polygons |37
Interior and Exterior Measures of Polygons |42
Classifying Triangles and the Triangle Inequality Theorem |50
The Quadrilateral Family |57
SELF TEST 2: Polygons |62
3. Indirect Measure
65
Pythagorean Theorem, Part 1 |65
Pythagorean Theorem, Part 2 |70
SELF TEST 3: Indirect Measure |74
4.Review
77
LIFEPAC Test is located in the
center of the booklet. Please
remove before starting the unit.
Section 1 |1
Measurement | Unit 6
Author: Glynlyon Staff
Editor:
Alan Christopherson, M.S.
Westover Studios Design Team:
Phillip Pettet, Creative Lead
Teresa Davis, DTP Lead
Nick Castro
Andi Graham
Jerry Wingo
804 N. 2nd Ave. E.
Rock Rapids, IA 51246-1759
© MMXIV by Alpha Omega Publications a division of
Glynlyon, Inc. All rights reserved. LIFEPAC is a registered
trademark of Alpha Omega Publications, Inc.
All trademarks and/or service marks referenced in this material are the property of their respective
owners. Alpha Omega Publications, Inc. makes no claim of ownership to any trademarks and/
or service marks other than their own and their affiliates, and makes no claim of affiliation to any
companies whose trademarks may be listed in this material, other than their own.
Some clip art images used in this curriculum are from Corel Corporation, 1600 Carling Avenue, Ottawa,
Ontario, Canada K1Z 8R7. These images are specifically for viewing purposes only, to enhance the
presentation of this educational material. Any duplication, resyndication, or redistribution for any
other purpose is strictly prohibited. Other images in this unit are © 2009 JupiterImages Corporation
2| Section 1
Unit 6 | Measurement
Measurement
Introduction
This unit covers basic geometric concepts associated with angles, parallel and
perpendicular lines, and circles. The properties of various polygons and their
angle measures are used to solve for missing angle measures. The special
relationship that exists between the sides and angles of right triangles, including
a proof of the Pythagorean theorem, is also discussed.
Objectives
Read these objectives. The objectives tell you what you will be able to do when
you have successfully completed this LIFEPAC. When you have finished this
LIFEPAC, you should be able to:
zz Classify and measure angles and lines.
zz Identify and find measures of angles created by transversals.
zz Identify parts of circles and their measures.
zz Classify polygons and find measures of their interior and exterior angles.
zz Classify triangles and use the triangle inequality theorem.
zz Classify quadrilaterals and the relationships among them.
zz Find side lengths of right triangles using the Pythagorean theorem.
Section 1 |3
Measurement | Unit 6
Survey the LIFEPAC. Ask yourself some questions about this study and write your questions here.
________________________________________________________________________________________________ ________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
________________________________________________________________________________________________
4| Section 1
Unit 6 | Measurement
1. Angle Measures and Circles
Classify and Measure Angles
Math is part of our everyday world in
everything from balancing checkbooks, to
shopping, to measuring a room for new
carpet. Math is also a key part of many of
the games we play. For instance, the game
of pool is nothing but a game of angles.
Knowing the correct angle needed to hit the
ball is essential to winning a game of pool.
One of the best skills you can have if you
are a pool player is being able to identify
the angle and angle measure needed to
sink a shot.
Objectives
z Identify angles by their measure.
z Classify
z Find
pairs of angles.
the measure of an angle.
Vocabulary
acute angle—an angle that measures more than 0° but less than 90°
adjacent angles—two angles that have a common vertex and side but are not
overlapping
complementary angles—two angles whose sum is 90°
obtuse angle—an angle that measures more than 90° but less than 180°
right angle—an angle that measures exactly 90°
straight angle—an angle that measures exactly 180°
supplementary angles—two angles whose sum is 180°
vertex—the point where two line segments, lines, or rays meet to form an angle
vertical angles—angles that are opposite from one another at the intersection of two
lines; vertical angles are congruent
Section 1 |5
Measurement | Unit 6
Classifying Angles
Angles are a part of our everyday life. Just
by looking around the room you are in, you
should be able to see hundreds of angles!
Each angle or pair of angles has a name
and set of properties to define it.
Complementary angles:
You may remember the terms acute angle,
right angle, obtuse angle, and straight angle.
An acute angle is an angle that measures
more than 0° but less than 90°. A right
angle measures exactly 90°, while an
obtuse angle measures more than 90° but
less than 180°. A straight angle, or straight
line, measures exactly 180°.
Supplementary angles are two angles
whose sum is 180°; the angles can either be
separate or adjacent. The following graphic
shows two adjacent, supplementary angles.
You can easily classify an angle as acute,
right, obtuse, or straight just by looking at
it. If you remember that a right angle is like
a corner on a piece of paper and a straight
angle is a straight line, you can tell what
type of angle you have without measuring
it. If the angle is smaller than the corner of
a piece of paper, you have an acute angle.
If it is larger than the corner of a piece of
paper but smaller than a line, then you
have an obtuse angle.
Let’s look at how to classify pairs of angles.
Adjacent angles are two angles that share a
common vertex, point, and side. They are
next to one another without overlapping.
Two types of angles that are often
adjacent are complementary angles and
supplementary angles.
Complementary angles are two angles
whose sum is 90°. The two angles can
either be adjacent angles or separate
angles. The following graphic shows two
adjacent, complementary angles.
6| Section 1
Supplementary angles:
Angles are also created when lines
intersect.
When the two lines intersected, they
created four angles. The four angles all
share a common vertex, where the lines
intersect. Some of the angles can also be
described as being adjacent. Angle 1 is
adjacent to both angle 4 and angle 2. Angle
2 is adjacent to angle 1 and angle 3. Angle 3
is adjacent to angles 2 and 4. Finally, angle
4 is adjacent to angles 3 and 1.
Unit 6 | Measurement
Within these four angles, there are 2 sets
of vertical angles. Vertical angles are angles
that are opposite one another but share a
vertex. Angles 1 and 3 are vertical angles,
because they have the same vertex and
are opposite one another. The other set of
vertical angles are angles 2 and 4. Again,
they have the same vertex and are opposite
one another.
Solution:
►
There are four sets of vertical angles:
•
1 and
3
•
2 and
4
•
5 and
7
•
6 and
8
Example:
Vertical angles are easily identifiable,
because they are found where lines
intersect. Look at the following picture.
Three lines intersect at one point. The
newly created angles are labeled for you.
Solution:
►
Let’s look at more sets of vertical angles.
See if you can identify them before reading
the answers.
Example:
There are 6 sets of vertical angles:
•
1 and
4
•
2 and
3
•
5 and
8
•
6 and
7
•
9 and
12
•
10 and
11
Measurements of Angles
Now that you know how to classify angles
and pairs of angles, we can look into how
to find the exact measure of an angle.
Remember that complementary angles
have a sum of 90°, and supplementary
angles have a measure of 180°. Look at the
following examples to see how to find the
measure of the angle.
Section 1 |7
Measurement | Unit 6
Example:
Supplementary
x + 39° = 180° angles sum to 180°.
Subtract 39° from
x + 39° - 39° = 180° - 39° both sides.
Complete the
x = 141° subtractions.
Solution:
►
►
The first thing you need to
determine is if the pair of angles are
complementary or supplementary.
The angles are complementary. We
can tell this because there is a right
angle symbol.
Complementary
Subtract 72 from
x + 72° - 72° = 90° - 72° both sides.
Complete the
x = 18° subtractions.
The second angle in the
complementary pair of angles
measures 18°.
Example:
Solution:
►
8| Section 1
Example:
We now know that the two angles
have a sum of 90°. We also know that
one of the angles measures x, while
the other angle measures 72°. We
have enough information to set up
an equation to find the value of x.
x + 72° = 90° angles sum to 90°.
►
Let’s move on to vertical angles. You know
that vertical angles are angles that are
opposite one another and share a vertex.
Vertical angles are also congruent, or equal
to one another. This means once you know
the measure of one, you automatically
know the measure of the other.
This time we have a pair of
supplementary angles, so we need to
set the equation equal to 180°.
►
►
1 = 82°
What are the measures of the other
angles?
Solution:
►
1 is a vertical angle with 3.
Therefore, we now know that 3
equals 82°.
►
We can easily find angles 2 and 4.
We know that they are also a set of
vertical angles. This means that all we
have to do is find the measure of one
angle, and we’ll know the measure of
the other.
►
If we look closely at the image, we
see that angle 1 and angle 2 are
supplementary, because they form a
straight angle. Since supplementary
angles equal 180°, angle 1 equals
82°, and angle 2 equals x, you can
now set up an equation.
Unit 6 | Measurement
Supplementary
Supplementary
x + 82° = 180° angles sum to
x + 82° - 82° = 180° - 82°
x = 98°
►
180°.
Subtract 82° from
both sides.
Complete the
subtractions.
Angle 2 measures 98°. That means
angle 4 also equals 98°.
x + 135° = 180° angles sum to
x + 135° - 135° = 180° - 135°
x = 45°
►
We now know that angle 1 equals
45°. Since angle 1 and angle 4 are
vertical angles, we can determine
that angle 4 is also equal to 45°.
►
Let’s move to the intersection that
creates angles 5, 6, 7, and 8. You
were told that 8 = 160°. That
means angle 5 also equals 160°,
because 8 and 5 are vertical angles.
You can see that angles 5 and 6 are
supplementary angles, so you know
their sum is equal to 180°.
Example:
►
2 = 135°
►
8 = 160°
►
11 = 65°
►
What are the measures of the other
angles?
Supplementary
x + 160° = 180° angles sum to
x + 160° - 160° = 180° - 160°
x = 20°
Solution:
►
►
The easiest way to solve this problem
is to use one point of intersection at
a time.
Let’s begin with the intersection that
creates angles 1, 2, 3, and 4. So far,
we know that 2 = 135°. We also
know that angles 2 and 3 are vertical
angles, which means that they are
equal. Finally, we can see that angles
1 and 2 are supplementary. That
means they have a sum of 180°.
180°.
Subtract 135°
from both sides.
Complete the
subtractions.
►
►
180°.
Subtract 160°
from both sides.
Complete the
subtractions.
If angle 6 equals 20°, then angle 7
also equals 20°.
Finally, let’s look at the intersection
that creates angles 9, 10, 11, and
12. You know that angle 11 equals
65°, so that means angle 9 also
equals 65°. Angles 9 and 10 are
supplementary to one another.
Supplementary
x + 65° = 180° angles sum to 180°.
Subtract 65° from
x + 65° - 65° = 180° - 65° both sides.
Complete the
x = 115° subtractions.
►
Angle 10 measures 115°, as well as
angle 12.
Section 1 |9
Measurement | Unit 6
Let’s Review
Before going on to the practice problems,
make sure you understand the main points
of this lesson.
„ Identify
the types of angles.
„ Classify
pairs of angles.
„ Find
the measure of angles.
Complete the following activities.
1.1
If angle 1 measures 98° and angle 2 measures 72°, then the two angles are
supplementary.
True
{
False
{
1.2
Complementary angles have a sum of 90°.
True
{
False
{
1.3
All pairs of vertical angles are equal.
True
{
False
{
1.4
Adjacent angles are always supplementary.
True
{
False
{
1.5
If angle 1 and angle 5 are vertical angles and angle 1 equals 55°, then angle 5 will
equal _____.
25°
55°
125°
can’t be
…
…
…
…
determined
1.6
If each of two complementary angles has the same measure, then each angle will
equal _____.
180°
90°
45°
22.5°
…
…
…
…
10| Section 1
Unit 6 | Measurement
1.7
In the following diagram, angle 7 equals 61°.
What is the measure of angle 8?
29°
…
61°
…
119°
…
129°
…
1.8
If two angles are supplementary and one of the angles measures 38°, what is the
measure of the larger angle?
62°
152°
52°
142°
…
…
…
…
Use this illustration for the following questions.
1.9
Angle 3 is equal to angle _____________ .
1.10 Angle 2 is equal to angle _____________ .
1.11 Angle 7 is equal to angle _____________ .
1.12 Angle 6 is equal to angle _____________ .
Section 1 |11
Measurement | Unit 6
Perpendicular and Parallel Lines, Part 1
Pictured is the Hearst Tower in New York.
The tower earned the 2006 Emporis
Skyscraper Award for the best skyscraper in
the world completed that year. It was also
the first “green” building in New York.
Take a closer look at the triangular framing
design on the outside of the building. Can
you identify the different types of angles
that are created? Can you also identify the
different types of lines that exist?
Objectives
z Identify lines as parallel, intersecting,
or perpendicular.
z Identify
a transversal and the angles
it creates.
z Find
the measure of angles created by a transversal.
Vocabulary
alternate exterior angles—two outside angles that lie on different sides of a transversal
alternate interior angles—two inside angles that lie on different sides of a transversal
corresponding angles—two angles in the same position on different lines
exterior angles—outside angles
interior angles—inside angles
intersecting lines—lines that share one point
parallel lines—lines that never cross one another and are the same distance apart at all
times
perpendicular lines—lines that intersect and create right angles
transversals—lines that intersect two or more lines to create angles
Classifying Lines
Every set of lines can be classified as
parallel, perpendicular, or intersecting. They
are often easy to tell apart, just by looking
at them.
Parallel lines are two lines that never
intersect. They are also always the same
12| Section 1
distance apart. Intersecting lines are lines
that cross one another at one point. Finally,
perpendicular lines are lines that cross one
another and create four right angles at the
point of intersection.
Unit 6 | Measurement
The transversal also creates two sets of
angles known as alternate exterior angles.
Alternate exterior angles are angles that
are on the opposite sides of the transversal,
opposite line, and on the outside of the
lines.
Sometimes, another line will cross a set of
lines in a different direction. This individual
line is called a transversal. A transversal
is a line that crosses two or more lines
and creates angles. Take a look at some
examples of transversals. The transversals
are the red lines in each picture. Notice
that the transversal can cut horizontally,
vertically, or diagonally.
The last two sets of angles created by the
transversal are alternate interior angles.
Alternate interior angles are angles that are
on the opposite sides of the transversal,
opposite line, and are on the inside of the
lines.
When a transversal cuts across a set of two
lines, it creates eight angles, four angles per
intersection. The angles created on the two
lines have a relationship with one another.
One type of angle that is created is called
corresponding angles. Corresponding angles
are angles that are in the same position but
on different lines. There are four sets of
corresponding angles.
When a transversal cuts across a set of
parallel lines, the angles that are created
have a special relationship. Corresponding
angles, alternate interior angles, and
alternate exterior angles are all congruent
within their own pairs.
Key point! The sets of angles only become
congruent when a transversal cuts across a
set of parallel lines!
Section 1 |13
Measurement | Unit 6
Example:
►
Identify the two sets of alternate
interior angles. Assume the lines are
parallel.
Solution:
Solution:
►
2 and
7
►
3 and
6
Example:
►
Identify the two sets of alternate
exterior angles. Assume the lines are
parallel.
►
1 and
3
►
2 and
4
►
5 and
7
►
6 and
8
Let’s look at some examples where we can
use congruency to find the measures of the
eight angles created by a transversal.
Example:
►
What are the measures of each
angle?
►
line a || line b
Solution:
►
1 and
7
►
2 and
8
Example:
►
Identify the four sets of
corresponding angles. Assume the
lines are parallel.
14| Section 1
►
2 = 70°
Solution:
►
Let’s begin by finding the value of
3. You know that angles 2 are 3
are vertical angles, and that vertical
angles are congruent. So, the
measure of angle 3 is 70°.
Unit 6 | Measurement
►
You also know that angle 1 and 2 are
supplementary, which means their
sum is equal to 180°. Since angle 2
measures 70°, then angle 1 must
measure 110°.
►
You can now use either vertical
angles or supplementary angles to
find the measure of angle 4. Either
way, angle 4 will measure 110°.
►
Take a look at what you have so far.
►
►
►
►
Now, let’s use the new angle
relationships you just learned about.
You can use corresponding angles,
alternate interior angles, or alternate
exterior angles to move from line
a to line b. Let’s use corresponding
angles this time.
Angle 1 is in the upper left of the
intersection on line a. Angle 5 is in
the upper left of the intersection
on line b. Because lines a and b are
parallel, you know that angle 1 and
angle 5 are congruent. So, angle 5
has a measure of 110°.
Now you can use vertical angles to
find that the measure of angle 8 is
also 110°.
Again, use supplementary angles to
find the measure of angle 6. Since
angle 5 measures 110°, then angle 6
must measure 70°.
►
Finally, use vertical angles to find
angle 7. Since angles 6 and 7 are
vertical, and angle 6 measures 70°,
then angle 7 also measures 70°.
►
Here is the completed diagram with
angle measures.
►
See if you can spot the four sets of
corresponding angles, the two sets of
alternate interior angles, and the two
sets of alternate exterior angles. Is
each set congruent?
Example:
►
Find the measure of each angle.
►
line x || line y
►
6 = 47°
Solution:
►
You know that angle 1 is a vertical
angle to angle 6; so, it also must have
a measure of 47°.
►
You also know that angle 1 and angle
2 are supplementary, so they have
Section 1 |15
Measurement | Unit 6
a sum of 180°. Since 47° of the 180°
is occupied by angle 1, that leaves
angle 2 to measure 133°.
►
Angle 5 is a vertical angle to angle 2;
so, that means it also has a measure
of 133°.
►
This time, let’s use alternate exterior
angles to move between the lines.
Angle 5 is an outside angle, so its
alternate exterior angle is angle 4.
Since the lines are parallel, these two
angles are congruent, giving angle 4
a measure of 133°.
►
Angle 4 is a vertical angle to angle
7; so, angle 7 also has a measure of
133°.
►
That leaves angles 3 and 8 to find.
Angle 3 corresponds to angle 1;
so that means they have the same
measure of 47°. Since angle 3 and
angle 8 are vertical, angle 8 also has
a measure of 47°.
►
Be Careful! Remember that transversals
always create sets of corresponding angles,
alternate interior angles, and alternate
exterior angles. However, the sets of angles
are only congruent when the transverse set
of lines is parallel.
Solution:
►
Let’s start with line m. Angles 1 and 4
are vertical angles; so, angle 4 must
also measure 61°. Angles 1 and 2
are supplementary angles; so, that
means that angle 2 measures 119°.
Finally, for line m, angles 2 and 3 are
vertical angles; so, angle 3 must also
measure 119°.
►
Even though corresponding
angles, alternate exterior angles,
and alternate interior angles exist
between the two lines, they are not
congruent, so we can’t equate the
measures. If you look back at the
original problem, you were given the
measure for angle 6.
►
Since angle 6 is a vertical angle with
angle 7, you know that the measure
of angle 7 is also 102°. Angle 6 and
angle 8 are supplementary angles;
so, that means angle 8 has to have a
measure of 78°. Angle 8 is a vertical
angle with angle 5, which means that
angle 5 also has a measure of 78°.
There is no one correct way to find the
measures of the angles, so don’t worry if
you and your friend have different ways of
finding the answers.
Let’s see what happens when the set of
lines being crossed by the transversal aren’t
parallel.
Example:
16| Section 1
Find the measures of each angle, if m
1 = 61° and m 6 = 102°.
Unit 6 | Measurement
Let’s Review
Before going on to the practice problems,
make sure you understand the main points
of this lesson.
„ Identify
„ Identify
„ Find
lines as parallel, intersecting, or
perpendicular.
creates.
a transversal and the angles it
„ Identify
the sets of angles created by a
transversal.
the measure of angles created by a
transversal.
Complete the following activities.
Use the following illustration for 1.13-1.15. Assume the lines are parallel.
1.13 Select all pairs of corresponding angles.
1 and 5
2 and 6
…
…
2 and 5
…
3 and 8
…
3 and 7
…
1.14 Select all pairs of alternate interior angles.
3 and 4
3 and 6
…
…
3 and 5
…
4 and 5
…
1.15 Select all pairs of alternate exterior angles.
1 and 2
1 and 8
…
…
1 and 7
…
2 and 7
…
4 and 7
…
4 and 8
…
1 and 6
…
none
…
4 and 6
…
5 and 6
…
none
…
2 and 8
…
7 and 8
…
none
…
Section 1 |17
Measurement | Unit 6
Use the following illustration for 1.16-1.17. Assume the lines are parallel.
1.16 Select all of the angles that have the same measure as angle 1.
… 2
… 4
… 6
… 3
… 5
… 7
… 8
can’t be
…
determined
1.17 Select all of the angles that have the same measure as angle 2.
… 1
… 4
… 6
… 3
… 5
… 7
1.18 The value of x that makes lines a and b parallel is ______°.
18| Section 1
… 8
can’t be
…
determined
Unit 6 | Measurement
1.19 The measure of angle 7 is ______°.
1.20 Line ______ is a transversal.
Use the following illustration for 1.21-1.22. Assume the lines are parallel.
1.21 Angle 4 corresponds with angle _________ .
1.22 Find the measure of each angle.
m 2 = _______ °.
m 6 = _______ °.
m 3 = _______ °.
m 7 = _______ °.
m 4 = _______ °.
m 8 = _______ °.
m 5 = _______ °.
Section 1 |19
Measurement | Unit 6
Perpendicular and Parallel Lines, Part 2
What is the only angle from which you
should deal with a problem?
Answer: Q ZKN-QFUST!
Can you figure out what the answer to the
question is?
Try using the secret decoder ring to help
figure it out. Use the outside circle for the
letter in the code. Replace the code letter
with the letter on the inner circle.
This lesson focuses on finding the measures
of angles when there are variables in the
angles. It is similar to being written in code,
except this code is in algebraic form with
variables.
Objectives
z Identify the relationships between angles created by a transversal across parallel
lines.
z Find
the measure of the angles created by a transversal across parallel lines.
Finding Angle Measures
The only angle from which you should deal
with a problem is a TRY-angle!
The study of writing in code is called
cryptography. Sometimes governments
send coded messages, and, sometimes,
you and your friends do, too. Have you ever
written a note in code, so that only you and
the person you are writing to know what it
says?
Previously, you learned about vertical
angles, complementary angles,
supplementary angles, and angles created
by transversals. We are going to continue
our study of these angles, but this time we
are going to have variables and variable
20| Section 1
expressions as our angle measures. Just
like any other time you see a variable in an
equation, you need to find the value of the
variable to find the angle measure.
Let’s begin by working with complementary
and supplementary angles. Remember that
complementary angles have a sum of 90°,
and supplementary angles have a sum of
180°.
Example:
Unit 6 | Measurement
►
Find the measure of the two angles.
►
Be careful—the value of 15° doesn’t
tell us either angle measure. It does,
however, help us, because we now
know the value of x. All we have to
do is plug 15° into the expression for
each angle.
Supplementary angles sum
►
4x = 4(15°) = 60°
Complete the addition on the
►
2x = 2(15°) = 30°
Solution:
►
We have a pair of supplementary
angles. So, we need to set the
equation equal to 180°.
x + 3x = 180° to 180°.
4x = 180° left.
=
Example:
Divide both sides by 4.
x = 45° Complete the division.
►
We know that x is 45°. That means
we can also find the other angle of 3x.
►
3x = 3(45°) = 135°
►
The two angle measures are 45° and
135°.
Example:
Solution:
►
We have a set of supplementary
angles, which sum to 180°.
Supplementary
2x + (5x + 33°) = 180° angles sum to 180°.
7x + 33° = 180° Add the variables.
►
Find the measure of each angle.
Solution:
►
Subtract 33° from
7x + 33° - 33° = 180° - 33° both sides.
=
This is a set of complementary
angles. So, the sum of the angles will
be equal to 90°.
Complementary angles sum
4x + 2x = 90° to 90°.
Complete the addition on the
6x = 90° left.
=
Divide both sides by 6.
x = 15° Complete the division.
Divide both sides
by 7.
Complete the
x = 21° division.
►
So far, we know that x = 21°. But, that
doesn’t tell us either angle measure.
To find the angle measures, we
need to plug 21° in for x in each
expression.
2x = 2(21°) = 42°
5x + 33° = 5(21°) + 33° = 105° + 33° = 138°
Section 1 |21
Measurement | Unit 6
►
The two angles measure 42° and
138°. You can check your work by
adding them together to make sure
you get a sum of 180°.
►
x + 34° = 17° + 34° = 51°
Example:
Now, let’s look at what happens when we
replace angle measures in vertical angles.
Remember that vertical angles are always
equal to one another.
Example:
►
Find the measure of the shaded
angle.
Solution:
►
►
Find the value of x and, then, the
measure of the vertical angles.
Supplementary
Solution:
►
Since vertical angles are equal in
measure, we need to put each angle
on either side of the equal sign.
►
3x = x + 34°
►
You solve this equation just as you
would solve any other equation, by
solving for x.
(4x + 25°) + (7x - 21°) = 180° angles sum to
11x + 4° = 180°
11x + 4° - 4° = 180° - 4°
11x = 176°
3x = x + 34° Vertical angles are congruent.
3x - x = x - x + 34° Subtract x from both sides.
2x = 34° Simplify.
=
x = 17°
=
Complete the division.
Now that you know the value of x,
plug that value into each angle’s
expression to make sure that they
are equal to one another.
►
3x = 3(17°) = 51°
180°.
Combine the
variables and
constants on the
left side.
Subtract 4° from
both sides.
Complete the
subtraction.
Divide both sides
by 11.
Complete the
x = 16° division.
Divide both sides by 2.
►
22| Section 1
We see that the two angles for which
we have values are supplementary
angles. So, we can add them together
to get a sum of 180°.
►
Now, plug that value into the
expression 7x - 21°, because it is a
vertical angle to the shaded angle.
►
7x - 21° =
7(16°) - 21° =
112° - 21° = 91°
►
The measure of the shaded angle is
91°.
Unit 6 | Measurement
Finally, let’s take a look at angles that are
created by a transversal.
Example:
►
►
►
m 5 = 3x = 3(31°) = 93°
►
Angle 5 measures 93°.
►
m 6 = 2x + 25° =
2(31°) + 25° =
62° + 25° = 87°
►
Angle 6 measures 87°.
►
Now you can use vertical angles to
finish this intersection. Angle 2 is
a vertical angle with angle 5; so, it
measures 93°. Angle 1 is a vertical
angle with angle 6; so, it has a
measure of 87°.
►
If you use corresponding angles,
angle 7 corresponds with angle 5; so,
angle 7 also measures 93°. If angle
7 measures 93°, then angle 4 also
measures 93°.
►
Angle 8 corresponds to angle 6; so, it
has a measure of 87°. Because angle
3 is a vertical angle with angle 8, it
too has a measure of 87°.
Find the measure of each angle.
You are given that m 5 = 3x and m
6 = 2x + 25.
Solution:
►
Angles 5 and 6 are supplementary
angles; so, we know they have a
sum of 180°. Therefore, we begin
by adding their values and setting it
equal to 180°.
Supplementary
3x + (2x + 25°) = 180° angles sum to 180°.
5x + 25° = 180°
5x + 25° - 25° = 180° - 25°
5x = 155°
=
Combine the
variables on the left
side.
Subtract 25° from
both sides.
Complete the
subtraction.
Divide both sides
by 5.
Complete the
x = 31° division.
►
Let’s Review
Before going on to the practice problems,
make sure you understand the main points
of this lesson.
„ A
transversal across parallel lines
creates corresponding, alternate
interior, and alternate exterior angles.
„ You
can use congruence among angles,
complementary relationships, and
supplementary relationships to calculate
the measures of unknown angles.
This just tells us the value of x, not an
angle measure. We need to plug 31°
into each of the expressions to get
the angle measures.
Section 1 |23
Measurement | Unit 6
Complete the following activities.
1.23 Use the following diagram to answer the questions.
Given A || B.
The value of x is ________ .
The measure of
5 is ________°.
The measure of
1 is ________°.
The measure of
6 is ________°.
The measure of
2 is ________°.
The measure of
7 is ________°.
The measure of
3 is ________°.
The measure of
8 is ________°.
The measure of
4 is ________°.
Angles 2 and 3 are _________________
angles.
1.24 Use the diagram and information to match the following angles with their correct
measures. Given Z = 2b + 6 and Y = 3b - 1.
_________
_________
_________
_________
24| Section 1
W
X
Y
Z
130°
50°
140°
40°
Unit 6 | Measurement
Circles
Think back to kindergarten when you first
began to learn about shapes. You knew that
the triangle had three sides, the square had
four sides, and that the circle was the round
shape without any sides. You could even
identify things around the classroom that
were those shapes.
But, do you know that there is more to
these shapes, especially the circle, then
what they taught you in kindergarten? This
lesson will focus on the different parts of a
circle.
Objectives
z Identify the parts of a circle.
z Classify
angles of circles.
z Find
the measures of arcs and angles
of circles.
Vocabulary
arc—part of the circumference of a circle between two points
central angle—an angle that has the center of a circle as its vertex
chord—a line segment that connects two points on a circle
circle—a figure with all of its points the same distance from the center
circumference of a circle—the length around the exterior of a circle; measured in
degrees in terms of an arc
diameter—a line segment that goes through the center of a circle to connect two points
on the circle
inscribed angle—an angle in a circle with a vertex on the circle and sides that are chords
of the circle
intercepted arc—the part of the circumference of a circle that is bound by two endpoints
of a chord, radius, secant line or tangent line
major arc—an arc with a measure greater than 180 degrees
minor arc—an arc with a measure less than 180 degrees
radius—a line segment from the center of a circle to any point on the circle
secant—a line that goes through the circle at two distinct points on the circumference of
a circle; the distance between the two points is considered a chord
Section 1 |25
Measurement | Unit 6
semi-circle—an arc with a measure of 180 degrees
tangent—a line that touches the exterior part of the circumference of a circle at exactly
one point
You know that a circle is a round shape that
doesn’t have any sides, but do you know its
real definition?
A circle is a figure that has all of its points
the same distance from the center. A circle
is named by its center point. Look at the
following figure. Notice the center dot
labeled A. Each of the points that surround
this point is the same distance as the other
points.
Sometimes, there are line segments that
connect two points on the circle. This line
segment might not always go through
the center of the circle. If a line segment
connects two points on the circle, then it is
called a chord. Look at the picture of some
examples of chords. Chords do not all have
the same measure because they can have
different lengths. A chord can sometimes
be a diameter but a chord can never be a
radius.
Now that you know what a circle really is,
let’s look into the parts of the circle.
Inside the Circle
The radius is a line segment that connects
the center of a circle to any point on the
circle. It doesn’t matter where you connect
the center of the circle to the edge to form
the radius, because all the points that make
up the circle are the same distance from
the center. Therefore, every line segment
drawn from the center of the circle to the
edge of the circle has the same measure for
that circle.
26| Section 1
A diameter is a special type of chord. Like
all other chords it connects two points on
a circle. But, for a chord to be a diameter,
it must go through the center of the circle.
Another way to think of a diameter is two
Unit 6 | Measurement
radiuses put together to form a straight
line. The measure of a diameter is twice
that of the radius.
travels through the inside of a circle, while
the secant does.
Did you know! A radius is half the measure
of the diameter. So, r =
d and d = 2r.
Angles of a Circle
Outside the Circle
So far, we have looked at line segments
inside the circle, but what happens when
there are lines, too? The first line we will
look at is called a secant. A secant is made
up of a chord but it continues past the edge
of the circle.
There is another line that is found
completely on the outside of the circle. A
tangent is a line that touches the circle at
exactly one point. It is easy to tell a tangent
from a secant, because a tangent line never
You know that an angle is two rays or line
segments that share a common vertex. The
same is true for angles found in a circle. The
first kind of angle we are going to look at is
called a central angle. A central angle is an
angle that has the center point of the circle
as its vertex. Each leg of a central angle is
a radius of the circle. Look at the example
where the central angle is ∠BAD.
Central Angle
There is another type of angle found within
a circle. It is called an inscribed angle. An
inscribed angle is an angle in a circle with
a vertex on the circle and with sides that
are chords of the circle. This is a little
different from the central angle, because
Section 1 |27
Measurement | Unit 6
an inscribed angle’s vertex is not the center
of the circle but a point on the circle. Look
closely at the following picture; do you
see the difference between the inscribed
angle, ∠XYZ, and the central angle shown
previously?
Inscribed Angle
Arcs of a Circle
What is an arc? An arc is part of a circle
between two points on the circle. Look at
the following examples.
Reminder! A circle has a measure of 360°.
Arcs can be all different sizes. Similar to
angles of triangles, we can classify arcs by
their measure.
„ A
minor arc is an arc that has a measure
less than 180°.
This might help! If an arc is half of the circle,
then it is a semi-circle. If the arc is less than
half of the circle, then it is a minor arc. If the
arc is more than half of the circle, then it is a
major arc.
„ A
semi-circle is an arc with a measure of
exactly 180°.
Circle A has the following arcs:
28| Section 1
Unit 6 | Measurement
„ An
arc that has a measure greater than
180° is called a major arc.
„ There
is also a special arc called an
intercepted arc. An intercepted arc is an
arc that is created by an inscribed angle.
Remember that an inscribed angle is
an angle whose vertex is on the circle.
Thus, the intercepted arc is the part of
the circle between the legs of this angle.
Look at the following picture to help you
visualize an intercepted arc.
intercepted arc created by the angle. For
example, if the measure of a central angle
is 79°, then the measure of the arc created
is also 79°. It also works the other way. If
you know the measure of the arc created by
a central angle is 17°, then you know that
the measure of the central angle is also 17°.
An inscribed angle is an angle whose vertex
is on the circle and whose legs are chords
of the circle. The intercepted arc created by
the inscribed angle is two times larger than
the measure of the inscribed angle. Let’s
look at some examples.
Example:
►
What is the measure of arc BD?
Solution:
►
Phew! That was a lot of information to
digest all at once. Before learning more
about the measures of the angles and arcs
of a circle, consider reviewing the definition
and picture for each part of a circle.
The first step is to determine if the
angle that creates the arc is a central
angle or an inscribed angle. The
angle, ∠BAD, is a central angle. That
means the arc will have the same
measure. The angle measures 63°;
so, arc BD also measures 63°.
Measures of Angles and Arcs
Central angles are the angles whose vertex
is the center of the circle and the legs
of the angle are radii. The measure of a
central angle is equal to the measure of the
Section 1 |29
Measurement | Unit 6
Example:
►
What is the measure of arc HT?
Solution:
►
►
The arc is formed from an inscribed
angle. So, this time the measure of
the angle is half of the arc. To find
the measure of the arc, multiply the
measure of the angle by 2.
74°(2) = 148°
Example:
►
What is the measure of arc RB?
Solution:
►
►
This time, you know the angle has
a measure of 62°, and that it is an
inscribed angle. This means that
the arc measure is twice the angle’s
measure.
62°(2) = 124°
30| Section 1
Example:
►
What is the value of x?
Solution:
►
Since the arc is created by a central
angle, we can set the two measures
equal to one another.
►
5x + 9 = 144°
►
5x = 135°
►
x = 27°
Example:
►
What is the measure of x?
Solution:
►
We have an intercepted arc and an
inscribed angle. That means that we
need to multiply the angle measure
by 2 and, then, set it equal to the arc
measure.
Unit 6 | Measurement
►
2(3x + 10°) = 104°
►
6x + 20° = 104°
►
6x = 84°
►
x = 14°
Be sure that you understand the
relationships that exist between the
different angles and the arcs that are
created by them. A central angle and its
arc have the same measure, while an
inscribed angle has half the measure of the
intercepted arc.
Let’s Review
Before going on to the practice problems,
make sure you understand the main points
of this lesson.
„ The
parts of a circle include the center,
equidistant points, central angles,
inscribed angles, arcs, intercepted arcs
and chords.
„ The
measure of a central angle is equal
to the measure of its intercepted arc.
„ The
measure of an inscribed angle is
half the measure of its intercepted arc.
„ The
measure of an intercepted arc is
twice the measure of its inscribed angle.
Complete the following activities.
1.25 A radius is _____________________ the diameter.
1.26 A semi-circle measures _____________________ degrees.
1.27 A _____________________ angle has the same measure as its arc.
1.28 A circle is named by its _____________________ .
1.29 A diameter is also a _____________________ .
1.30 If an inscribed angle measures 67°, how would you find the intercepted arc?
Set the angle and arc measures
Multiply the angle measure by
…
…
equal.
two.
Divide the angle measure by two.
…
1.31 A line that intersects a circle at exactly one point is called a _____.
chord
tangent
secant
…
…
…
Section 1 |31
Measurement | Unit 6
1.32 If an arc measures 179°, what kind of arc is it?
minor arc
major arc
semi-circle
…
…
…
1.33 If the measure of a central angle is 28°, then to find the measure of its arc, you
would need to do which of the following?
Nothing, they are equal.
Divide the angle measure by 2.
…
…
Multiply the angle measure by 2.
…
1.34 A chord is a diameter.
always
sometimes
…
…
never
…
1.35 If the diameter of the circle is 11 inches, then the radius will be _____ inches.
22
11
5.5
…
…
…
1.36 If the measure of an intercepted arc is 118°, and the measure of the inscribed angle
is 2x - 1, what is the equation to find the value of x?
2x - 1 = 118°
2x - 1 = 2(118°)
…
…
2(2x - 1) = 118°
…
1.37 The measure of the arc created by a central angle is 57°. What type of arc is this?
minor arc
major arc
semi-circle
…
…
…
1.38 The measure of an inscribed angle is 110°. What is the measure of the intercepted
arc?
55°
110°
220°
…
…
…
1.39 A secant line contains a chord.
always
sometimes
…
…
never
…
Review the material in this section in preparation for the Self Test. The Self Test
will check your mastery of this particular section. The items missed on this Self Test will
indicate specific area where restudy is needed for mastery.
32| Section 1
Unit 6 | Measurement
SELF TEST 1: Angle Measures and Circles
Complete the following activities (6 points, each numbered activity).
1.01 Match each picture with its correct label.
complementary angles
minor arc
acute angle
inscribed angle
straight angle
secant
intersecting lines
chord
_________ _________ _________ _________ _________ _________ _________ _________ Section 1 |33
Measurement | Unit 6
1.02 If a central angle measures 87°, then its arc will measure _____.
half as much
the same
…
…
twice as much
…
1.03Given 1 = 123° and
parallel?
8 = 7z + 11. What is the value of z, if lines a and b are
…15°
16°
…
17°
…
1.04 What is the measure of x in the diagram?
13°
…
14°
…
118°
…
1.05 If an intercepted arc measures 124°, what is the measure of its inscribed angle?
the same, 124°
twice the measure, 248°
…
…
half the measure, 62°
…
1.06 Two angles are supplementary. One angle measures 2x, and the other angle
measures 3x - 15. Which equation can you use to find the measure of each angle?
2x + 3x - 15 = 90°
2x - 3x - 15 = 90°
…
…
2x - 3x - 15 = 180°
…
2x + 3x - 15 = 180°
…
1.07 If corresponding angles are on parallel lines, then their measure is the same _____.
always
sometimes
never
…
…
…
1.08 The measure of a central angle is 3x + 18, and the measure of its arc is 147°. Which
equation can you use to find the value of x?
2(3x + 18) = 147°
…
3x + 18 = 2(147°)
…
3x + 18 = 147°
…
34| Section 1
Unit 6 | Measurement
1.09 If two angles are complementary, then the sum of their angles is equal to _____.
each other
90°
180°
…
…
…
1.010 What is the measure of x?
43°
…
86°
…
172°
…
344°
…
1.011 If one angle of a set of alternate interior angles on parallel lines measures 77°, then
the other angle also equals 77° _____.
always
sometimes
never
…
…
…
1.012 In the graphic, one pair of vertical angles is ________ .
1.013 In the image, the corresponding angle to angle 1 is angle ____________ .
Section 1 |35
Measurement | Unit 6
1.014 If one angle of a set of supplementary angles measures 77°, then the other angle
measures _____.
13°
103°
…
…
77°
…
283°
…
1.015 If one angle of a set of vertical angles measures 63°, the sum of the vertical angles
is ________________ .
72
90
36| Section 1
SCORE
TEACHER
initials
date
MAT0806 – May ‘14 Printing
ISBN 978-0-7403-3184-8
804 N. 2nd Ave. E.
Rock Rapids, IA 51246-1759
9 780740 331848
800-622-3070
www.aop.com