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C HAPTER
Chapter 1. Triangle Classification
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Triangle Classification
Here you’ll learn how to classify triangles based on their angle and side measures.
What if you were given the angle measures and/or side lengths of a triangle? How would you describe the triangle
based on that information? After completing this Concept, you’ll be able to classify a triangle as right, obtuse, acute,
equiangular, scalene, isosceles, and/or equilateral.
Watch This
Watch this video beginning at around the 2:38 mark.
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James Sousa:Types of Triangles
Guidance
A triangle is any closed figure made by three line segments intersecting at their endpoints. Every triangle has three
vertices (the points where the segments meet), three sides (the segments), and three interior angles (formed at each
vertex). All of the following shapes are triangles.
The sum of the interior angles in a triangle is 180◦ . This is called the Triangle Sum Theorem and is discussed
further TriangleSumTheorem.
Angles can be classified by their size as acute, obtuse or right. In any triangle, two of the angles will always be
acute. The third angle can be acute, obtuse, or right. We classify each triangle by this angle.
Right Triangle: A triangle with one right angle.
Obtuse Triangle: A triangle with one obtuse angle.
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Acute Triangle: A triangle where all three angles are acute.
Equiangular Triangle: A triangle where all the angles are congruent.
You can also classify a triangle by its sides.
Scalene Triangle: A triangle where all three sides are different lengths.
Isosceles Triangle: A triangle with at least two congruent sides.
Equilateral Triangle: A triangle with three congruent sides.
Note that from the definitions, an equilateral triangle is also an isosceles triangle.
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Chapter 1. Triangle Classification
Example A
Which of the figures below are not triangles?
B is not a triangle because it has one curved side. D is not closed, so it is not a triangle either.
Example B
Which term best describes �RST below?
This triangle has one labeled obtuse angle of 92◦ . Triangles can have only one obtuse angle, so it is an obtuse
triangle.
Example C
Classify the triangle by its sides and angles.
We see that there are two congruent sides, so it is isosceles. By the angles, they all look acute. We say this is an
acute isosceles triangle.
Vocabulary
A triangle is any closed figure made by three line segments intersecting at their endpoints. Every triangle has three
vertices (the points where the segments meet), three sides (the segments), and three interior angles (formed at each
vertex). A right triangle is a triangle with one right angle. An obtuse triangle is a triangle with one obtuse angle. An
acute triangle is a triangle where all three angles are acute. An equiangular triangle is a triangle with all congruent
angles. A scalene triangle is a triangle where all three sides are different lengths. An isosceles triangle is a triangle
with at least two congruent sides. An equilateral triangle is a triangle with three congruent sides.
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Guided Practice
1. How many triangles are in the diagram below?
2. Classify the triangle by its sides and angles.
3. True or false: An equilateral triangle is equiangular.
Answers:
1. Start by counting the smallest triangles, 16.
Now count the triangles that are formed by 4 of the smaller triangles, 7.
Next, count the triangles that are formed by 9 of the smaller triangles, 3.
Finally, there is the one triangle formed by all 16 smaller triangles. Adding these numbers together, we get 16 + 7 +
3 + 1 = 27.
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Chapter 1. Triangle Classification
2. This triangle has a right angle and no sides are marked congruent. So, it is a right scalene triangle.
3. True. Equilateral triangles have interior angles that are all congruent so they are equiangular.
Practice
For questions 1-6, classify each triangle by its sides and by its angles.
1.
2.
3.
4.
5.
6.
7. Can you draw a triangle with a right angle and an obtuse angle? Why or why not?
8. In an isosceles triangle, can the angles opposite the congruent sides be obtuse?
For 9-10, determine if the statement is true or false.
9. Obtuse triangles can be isosceles.
10. A right triangle is acute.
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C HAPTER
2
Triangle Sum Theorem
Here you’ll learn how to use the Triangle Sum Theorem, which states that the sum of the angles in any triangle is
180◦ .
What if you knew that two of the angles in a triangle measured 55◦ ? How could you find the measure of the third
angle? After completing this Concept, you’ll be able to apply the Triangle Sum Theorem to solve problems like this
one.
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James Sousa:Animation of the Sum oftheInterior Anglesof a Triangle
Now watch this video.
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James Sousa:ProvingtheTriangle Sum Theorem
Guidance
The Triangle Sum Theorem says that the three interior angles of any triangle add up to 180◦ .
m� 1 + m� 2 + m� 3 = 180◦
.
Here is one proof of the Triangle Sum Theorem.
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Chapter 2. Triangle Sum Theorem
←
→
Given : �ABC with AD||BC
Prove : m� 1 + m� 2 + m� 3 = 180◦
TABLE 2.1:
Statement
←
→
1. �ABC with AD||BC
2. � 1 ∼
= � 4, � 2 ∼
=� 5
3. m� 1 = m� 4, m� 2 = m� 5
4. m� 4 + m� CAD = 180◦
5. m� 3 + m� 5 = m� CAD
6. m� 4 + m� 3 + m� 5 = 180◦
7. m� 1 + m� 3 + m� 2 = 180◦
Reason
Given
Alternate Interior Angles Theorem
∼
= angles have = measures
Linear Pair Postulate
Angle Addition Postulate
Substitution PoE
Substitution PoE
You can use the Triangle Sum Theorem to find missing angles in triangles.
Example A
What is m� T ?
We know that the three angles in the triangle must add up to 180◦ . To solve this problem, set up an equation and
substitute in the information you know.
m� M + m� A + m� T = 180◦
82◦ + 27◦ + m� T = 180◦
109◦ + m� T = 180◦
m� T = 71◦
Example B
What is the measure of each angle in an equiangular triangle?
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To solve, remember that �ABC is an equiangular triangle, so all three angles are equal. Write an equation.
m� A + m� B + m� C = 180◦
m� A + m� A + m� A = 180◦
3m�
A = 180
◦
Substitute, all angles are equal.
Combine like terms.
◦
m� A = 60
If m� A = 60◦ , then m� B = 60◦ and m� C = 60◦ .
Each angle in an equiangular triangle is 60◦ .
Example C
Find the measure of the missing angle.
We know that m� O = 41◦ and m� G = 90◦ because it is a right angle. Set up an equation like in Example A.
m� D + m� O + m� G = 180◦
m� D + 41◦ + 90◦ = 180◦
m� D + 41◦ = 90◦
m� D = 49◦
Vocabulary
A triangle is a three sided shape. All triangles have three interior angles, which are the inside angles connecting the
sides of the triangle.
Guided Practice
1. Determine m� 1 in this triangle:
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Chapter 2. Triangle Sum Theorem
2. Two interior angles of a triangle measure 50◦ and 70◦ . What is the third interior angle of the triangle?
3. Find the value of x and the measure of each angle.
Answers:
1. 72◦ + 65◦ + m� 1 = 180◦ .
Solve this equation and you find that m� 1 = 43◦ .
2. 50◦ + 70◦ + x = 180◦ .
Solve this equation and you find that the third angle is 60◦ .
3. All the angles add up to 180◦ .
(8x − 1)◦ + (3x + 9)◦ + (3x + 4)◦ = 180◦
(14x + 12)◦ = 180◦
14x = 168
x = 12
Substitute in 12 for x to find each angle.
[3(12) + 9]◦ = 45◦
[3(12) + 4]◦ = 40◦
[8(12) − 1]◦ = 95◦
Practice
Determine m� 1 in each triangle.
1.
2.
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3.
4.
5.
6.
7.
8. Two interior angles of a triangle measure 32◦ and 64◦ . What is the third interior angle of the triangle?
9. Two interior angles of a triangle measure 111◦ and 12◦ . What is the third interior angle of the triangle?
10. Two interior angles of a triangle measure 2◦ and 157◦ . What is the third interior angle of the triangle?
Find the value of x and the measure of each angle.
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Chapter 2. Triangle Sum Theorem
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14.
15.
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C HAPTER
3
Exterior Angles Theorems
Here you’ll learn what an exterior angle is as well as two theorems involving exterior angles: that the sum of the
exterior angles is always 360◦ and that in a triangle, an exterior angle is equal to the sum of its remote interior angles.
What if you knew that two of the exterior angles of a triangle measured 130◦ ? How could you find the measure of
the third exterior angle? After completing this Concept, you’ll be able to apply the Exterior Angle Sum Theorem to
solve problems like this one.
Watch This
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James Sousa:Introduction totheExterior Anglesof a Triangle
Then watch this video.
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James Sousa:Proof that the Sum of the Exterior Angles of a Triangleis 360 Degrees
Finally, watch this video.
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James Sousa:Proof of the Exterior Angles Theorem
Guidance
An Exterior Angle is the angle formed by one side of a polygon and the extension of the adjacent side.
In all polygons, there are two sets of exterior angles, one that goes around clockwise and the other goes around
counterclockwise.
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Chapter 3. Exterior Angles Theorems
Notice that the interior angle and its adjacent exterior angle form a linear pair and add up to 180◦ .
m� 1 + m� 2 = 180◦
There are two important theorems to know involving exterior angles: the Exterior Angle Sum Theorem and the
Exterior Angle Theorem.
The Exterior Angle Sum Theorem states that the exterior angles of any polygon will always add up to 360◦ .
m� 1 + m� 2 + m� 3 = 360◦
m� 4 + m� 5 + m� 6 = 360◦
.
The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of its remote interior
angles. (Remote Interior Angles are the two interior angles in a triangle that are not adjacent to the indicated
exterior angle.)
m� A + m� B = m� ACD
.
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Example A
Find the measure of � RQS.
Notice that 112◦ is an exterior angle of �RQS and is supplementary to � RQS.
Set up an equation to solve for the missing angle.
112◦ + m� RQS = 180◦
m� RQS = 68◦
Example B
Find the measures of the numbered interior and exterior angles in the triangle.
We know that m� 1 + 92◦ = 180◦ because they form a linear pair. So, m� 1 = 88◦ .
Similarly, m� 2 + 123◦ = 180◦ because they form a linear pair. So, m� 2 = 57◦ .
We also know that the three interior angles must add up to 180◦ by the Triangle Sum Theorem.
m� 1 + m� 2 + m� 3 = 180◦
◦
by the Triangle Sum Theorem.
◦
88 + 57 + m� 3 = 180
m� 3 = 35◦
Lastly, m� 3 + m� 4 = 180◦
◦
35
+ m�
m�
4 = 180
4 = 145◦
Example C
What is the value of p in the triangle below?
14
◦
because they form a linear pair.
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Chapter 3. Exterior Angles Theorems
First, we need to find the missing exterior angle, which we will call x. Set up an equation using the Exterior Angle
Sum Theorem.
130◦ + 110◦ + x = 360◦
x = 360◦ − 130◦ − 110◦
x = 120◦
x and p add up to 180◦ because they are a linear pair.
x + p = 180◦
120◦ + p = 180◦
p = 60◦
Vocabulary
Interior Angles are the angles on the inside of a polygon while Exterior Angles are the angles on the outside of a
polygon. Remote Interior Angles are the two angles in a triangle that are not adjacent to the indicated exterior angle.
Two angles that make a straight line form a Linear Pair and thus add up to 180◦ . The Triangle Sum Theorem states
that the three interior angles of any triangle will always add up to 180◦ .
Guided Practice
1. Find m� C.
2. Two interior angles of a triangle are 40◦ and 73◦ . What are the measures of the three exterior angles of the
triangle?
3. Find the value of x and the measure of each angle.
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Answers:
1. Using the Exterior Angle Theorem
m� C + 16◦ = 121◦
m� C = 105◦
If you forget the Exterior Angle Theorem, you can do this problem just like Example C.
2. Remember that every interior angle forms a linear pair (adds up to 180◦ ) with an exterior angle. So, since one
of the interior angles is 40◦ that means that one of the exterior angles is 140◦ (because 40 + 140 = 180). Similarly,
since another one of the interior angles is 73◦ , one of the exterior angles must be 107◦ . The third interior angle is
not given to us, but we could figure it out using the Triangle Sum Theorem. We can also use the Exterior Angle
Sum Theorem. If two of the exterior angles are 140◦ and 107◦ , then the third Exterior Angle must be 113◦ since
140 + 107 + 113 = 360.
So, the measures of the three exterior angles are 140, 107 and 113.
3. Set up an equation using the Exterior Angle Theorem.
(4x + 2)◦ + (2x − 9)◦ = (5x + 13)◦
↑
�
remote interior angles
↑
exterior angle
◦
(6x − 7) = (5x + 13)◦
x = 20
Substitute in 20 for x to find each angle.
[4(20) + 2]◦ = 82◦
[2(20) − 9]◦ = 31◦
Practice
Determine m� 1.
1.
2.
16
Exterior angle: [5(20) + 13]◦ = 113◦
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Chapter 3. Exterior Angles Theorems
3.
4.
5.
6.
Use the following picture for the next three problems:
7. What is m� 1 + m� 2 + m� 3?
8. What is m� 4 + m� 5 + m� 6?
9. What is m� 7 + m� 8 + m� 9?
Solve for x.
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12.
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C HAPTER
Chapter 4. Third Angle Theorem
4
Third Angle Theorem
Here you’ll learn the Third Angle Theorem: If two triangles have two pairs of angles that are congruent, then the
third pair of angles will be congruent.
What if you were given �FGH and �XY Z and you were told that
F∼
=� X
�
and
�
G∼
=� Y
? What conclusion could you draw about � H and � Z? After completing this Concept, you’ll be able to make such a
conclusion.
Guidance
If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also
congruent. This is called the Third Angle Theorem.
If � A ∼
= � D and � B ∼
= � E, then � C ∼
= � F.
Example A
Determine the measure of the missing angles.
From the Third Angle Theorem, we know � C ∼
= � F. From the Triangle Sum Theorem, we know that the sum of the
◦
interior angles in each triangle is 180 .
m� A + m� B + m� C = 180◦
m� D + m� B + m� C = 180◦
42◦ + 83◦ + m� C = 180◦
m� C = 55◦ = m� F
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Example B
Explain why the Third Angle Theorem works.
The Third Angle Theorem is really like an extension of the Triangle Sum Theorem. Once you know two angles in
a triangle, you automatically know the third because of the Triangle Sum Theorem. This means that if you have
two triangles with two pairs of angles congruent between them, when you use the Triangle Sum Theorem on each
triangle to come up with the third angle you will get the same answer both times. Therefore, the third pair of angles
must also be congruent.
Example C
Determine the measure of all the angles in the triangle:
First we can see that m� DCA = 15◦ . This means that m� BAC = 15◦ also because they are alternate interior angles.
m� ABC = 153◦ was given. This means by the Triangle Sum Theorem that m� BCA = 12◦ . This means that m� CAD =
12◦ also because they are alternate interior angles. Finally, m� ADC = 153◦ by the Triangle Sum Theorem.
Vocabulary
Two figures are congruent if they have exactly the same size and shape. Two triangles are congruent if the three
corresponding angles and sides are congruent. The Triangle Sum Theorem states that the measure of the three
interior angles of any triangle will add up to 180◦ .
Guided Practice
Determine the measure of all the angles in the each triangle.
1.
2.
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Chapter 4. Third Angle Theorem
3.
Answers:
1. m� A = 86, m� C = 42 and by the Triangle Sum Theorem m� B = 52.
m� Y = 42, m� X = 86 and by the Triangle Sum Theorem, m� Z = 52.
2. m� C = m� A = m� Y = m� Z = 35. By the Triangle Sum Theorem m� B = m� X = 110.
3. m� A = 28, m� ABE = 90 and by the Triangle Sum Theorem, m� E = 62. m� D = m� E = 62 because they are
alternate interior angles and the lines are parallel. m� C = m� A = 28 because they are alternate interior angles and
the lines are parallel. m� DBC = m� ABE = 90 because they are vertical angles.
Practice
Determine the measures of the unknown angles.
1.
2.
�
�
XY Z
ZXY
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3.
4.
�
LNM
MLN
5.
6.
7.
�
CED
GFH
FHG
8.
9.
10.
11.
�
22
�
�
�
�
�
�
ACB
HIJ
HJI
IHJ
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12.
13.
14.
15.
�
�
�
�
Chapter 4. Third Angle Theorem
RQS
SRQ
T SU
TUS
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