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C HAPTER
1
1
Angle Classification
Here you’ll learn how to classify angles based on their angle measure.
What if you were given the degree measure of an angle? How would you describe that angle based on its size? After
completing this Concept, you’ll be able to classify an angle as acute, right, obtuse, or straight.
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Guidance
Angles can be grouped into four different categories.
Straight Angle: An angle that measures exactly 180◦ .
Acute Angles: Angles that measure between 0◦ and up to but not including 90◦ .
Obtuse Angles: Angles that measure more than 90◦ but less than 180◦ .
Right Angle: An angle that measures exactly 90◦ .
Chapter 1. Angle Classification
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This half-square marks right, or 90◦ , angles. When two lines intersect to form four right angles, the lines are
perpendicular. The symbol for perpendicular is ⊥.
Even though all four angles are 90◦ , only one needs to be marked with the half-square. l ⊥ m is read l is perpendicular
to line m.
Example A
What type of angle is 84◦ ?
84◦ is less than 90◦ , so it is acute.
Example B
Name the angle and determine what type of angle it is.
The vertex is U. So, the angle can be � TUV or � VUT . To determine what type of angle it is, compare it to a right
angle.
Because it opens wider than a right angle and is less than a straight angle, it is obtuse.
Example C
What type of angle is 165◦ ?
165◦ is greater than 90◦ , but less than 180◦ , so it is obtuse.
Vocabulary
A straight angle is an angle that measures exactly 180◦ . Acute angles are angles that measure between 0◦ and up to
but not including 90◦ . Obtuse angles are angles that measure more than 90◦ but less than 180◦ . A right angle is an
angle that measures exactly 90◦ . When two lines intersect to form four right angles, the lines are perpendicular.
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Guided Practice
Name each type of angle:
1. 90◦
2. 67◦
3. 180◦
Answers
1. Right
2. Acute
3. Straight
Interactive Practice
Practice
For exercises 1-4, determine if the statement is true or false.
1.
2.
3.
4.
Two angles always add up to be greater than 90◦ .
180◦ is an obtuse angle.
180◦ is a straight angle.
Two perpendicular lines intersect to form four right angles.
For exercises 5-10, state what type of angle it is.
5.
6.
7.
8.
9.
10.
55◦
92◦
178◦
5◦
120◦
73◦
In exercises 11-15, use the following information: Q is in the interior of � ROS. S is in the interior of � QOP. P is in
the interior of � SOT . S is in the interior of � ROT and m� ROT = 160◦ , m� SOT = 100◦ , and m� ROQ = m� QOS =
m� POT .
11.
12.
13.
14.
15.
Make a sketch.
Find m� QOP.
Find m� QOT .
Find m� ROQ.
Find m� SOP.
Chapter 1. Angle Classification
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C HAPTER
2
Congruent Angles and
Angle Bisectors
Here you’ll learn how to find unknown values using the definitions of angle congruency and angle bisector.
What if you were told that a line segment divides an angle in half? How would you find the measures of the two
new angles formed by that segment? After completing this Concept, you’ll be able to use the definitions of angle
congruency and angle bisector to find such angle measures.
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Then watch the first part of this video.
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Guidance
When two geometric figures have the same shape and size (or the same angle measure in the case of angles) they are
said to be congruent.
TABLE 2.1:
Label It
� ABC ∼
= � DEF
Say It
Angle ABC is congruent to angle DEF.
If two angles are congruent, then they are also equal. To label equal angles we use angle markings, as shown below:
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An angle bisector is a line, or a portion of a line, that divides an angle into two congruent angles, each having a
measure exactly half of the original angle. Every angle has exactly one angle bisector.
In the picture above, BD is the angle bisector of � ABC, so � ABD ∼
= � DBC and m� ABD = 12 m� ABC.
Example A
Write all equal angle statements.
m� ADB = m� BDC = m� FDE = 45◦
m� ADF = m� ADC = 90◦
Example B
What is the measure of each angle?
Chapter 2. Congruent Angles and Angle Bisectors
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From the picture, we see that the angles are equal.
Set the angles equal to each other and solve.
(5x + 7)◦ = (3x + 23)◦
(2x)◦ = 16◦
x=8
To find the measure of � ABC, plug in x = 8 to (5x + 7)◦ → (5(8) + 7)◦ = (40 + 7)◦ = 47◦ . Because m� ABC =
m� XY Z, m� XY Z = 47◦ too.
Example C
Is OP the angle bisector of � SOT ?
Yes, OP is the angle bisector of � SOT from the markings in the picture.
Vocabulary
When two geometric figures have the same shape and size then they are congruent. An angle bisector is a line or
portion of a line that divides an angle into two congruent angles, each having a measure exactly half of the original
angle.
Guided Practice
For exercises 1 and 2, copy the figure below and label it with the following information:
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1. � A ∼
=� C
2. � B ∼
=� D
3. Use algebra to determine the value of d:
Answers:
1. You should have corresponding markings on � A and � C.
2. You should have corresponding markings on � B and � D (that look different from the markings you made in #1).
3. The square marking means it is a 90◦ angle, so the two angles are congruent. Set up an equation and solve:
7d − 1 = 2d + 14
5d = 15
d=3
Interactive Practice
Practice
For 1-4, use the following picture to answer the questions.
1.
2.
3.
4.
What is the angle bisector of � T PR?
What is m� QPR?
What is m� T PS?
What is m� QPV ?
For 5-6, use algebra to determine the value of variable in each problem.
5.
Chapter 2. Congruent Angles and Angle Bisectors
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6.
For 7-10, decide if the statement is true or false.
7.
8.
9.
10.
Every angle has exactly one angle bisector.
Any marking on an angle means that the angle is 90◦ .
An angle bisector divides an angle into three congruent angles.
Congruent angles have the same measure.
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C HAPTER
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3
Angle Measurement
Here you’ll learn how to measure an angle with a protractor and how to apply the Angle Addition Postulate to find
unknown values.
What if you were given the measure of an angle and two unknown quantities that make up that angle? How would
you find the values of those quantities? After completing this Concept, you’ll be able to use the Angle Addition
Postulate to evaluate such quantities.
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MEDIA
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http://www.youtube.com/watch?v=N3I6OiO5mKI
Then look at the first part of this video.
MEDIA
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http://www.youtube.com/watch?v=7iBc5bJdanI
Guidance
An angle is formed when two rays have the same endpoint. The vertex is the common endpoint of the two rays that
form an angle. The sides are the two rays that form an angle.
TABLE 3.1:
Label It
� ABC
� CBA
Say It
Angle ABC
Angle CBA
Chapter 3. Angle Measurement
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−
→
−
→
The vertex is B and the sides are BA and BC. Always use three letters to name an angle, � SIDE-VERTEX-SIDE.
Angles are measured with something called a protractor. A protractor is a measuring device that measures how
“open” an angle is. Angles are measured in degrees and are labeled with a ◦ symbol. For now, angles are always
positive.
There are two sets of measurements, one starting on the left and the other on the right side of the protractor. Both go
around from 0◦ to 180◦ . When measuring angles, you can line up one side with 0◦ , and see where the other side hits
the protractor. The vertex lines up in the middle of the bottom line.
Note that if you don’t line up one side with 0◦ , the angle’s measure will be the difference of the degrees where the
sides of the angle intersect the protractor.
Sometimes you will want to draw an angle that is a specific number of degrees. Follow the steps below to draw a
50◦ angle with a protractor:
1. Start by drawing a horizontal line across the page, 2 in long.
2. Place an endpoint at the left side of your line.
3. Place the protractor on this point, such that the line passes through the 0◦ mark on the protractor and the endpoint
is at the center. Mark 50◦ on the appropriate scale.
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4. Remove the protractor and connect the vertex and the 50◦ mark.
This process can be used to draw any angle between 0◦ and 180◦ . See http://www.mathsisfun.com/geometry/protr
actor-using.html for an animation of this.
When two smaller angles form to make a larger angle, the sum of the measures of the smaller angles will equal the
measure of the larger angle. This is called the Angle Addition Postulate. So, if B is on the interior of � ADC, then
m� ADC = m� ADB + m� BDC
.
Example A
How many angles are in the picture below? Label each one.
Chapter 3. Angle Measurement
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There are three angles with vertex U. It might be easier to see them all if we separate them.
So, the three angles can be labeled, � XUY (or � YUX), � YUZ (or � ZUY ), and � XUZ (or � ZUX).
Example B
Measure the three angles from Example 1, using a protractor.
Just like in Example A, it might be easier to measure these three angles if we separate them.
With measurement, we put an m in front of the � sign to indicate measure. So, m� XUY = 84◦ , m� YUZ = 42◦ and
m� XUZ = 126◦ .
Example C
What is the measure of the angle shown below?
This angle is lined up with 0◦ , so where the second side intersects the protractor is the angle measure, which is 50◦ .
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Vocabulary
An angle is formed when two rays have the same endpoint. The vertex is the common endpoint of the two rays that
form an angle. The sides are the two rays that form an angle. Angles are measured with a protractor.
Guided Practice
1. What is the measure of the angle shown below?
2. Use a protractor to measure � RST below.
3. What is m� QRT in the diagram below?
Answers
1. This angle is not lined up with 0◦ , so use subtraction to find its measure. It does not matter which scale you use.
Inner scale: 140◦ − 15◦ = 125◦
Outer scale: 165◦ − 40◦ = 125◦
2. Lining up one side with 0◦ on the protractor, the other side hits 100◦ .
3. Using the Angle Addition Postulate, m� QRT = 15◦ + 30◦ = 45◦ .
Interactive Practice
Practice
1. What is m� LMN if m� LMO = 85◦ and m� NMO = 53◦ ?
Chapter 3. Angle Measurement
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2. If m� ABD = 100◦ , find x.
For questions 3-6, determine if the statement is true or false.
3.
4.
5.
6.
For an angle � ABC,C is the vertex.
For an angle � ABC, AB and BC are the sides.
The m in front of m� ABC means measure.
The Angle Addition Postulate says that an angle is equal to the sum of the smaller angles around it.
For 7-12, draw the angle with the given degree, using a protractor and a ruler.
7.
8.
9.
10.
11.
12.
55◦
92◦
178◦
5◦
120◦
73◦
For 13-16, use a protractor to determine the measure of each angle.
13.
14.
15.
16.
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Solve for x.
17. m� ADC = 56◦
Chapter 3. Angle Measurement