Triangle Angle­Sum Theorem
3.4 Triangle Angle Theorems
The sum of the measures of the angles of a triangle is 180.
Ex) m∠A + m∠B + m∠C = 180.
I CAN...
Apply the Triangle Angle­Sum Theorem and Exterior Angle Theorem
B
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Triangle Angle­Sum Theorem Proof C
A
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Triangle Angle­Sum Theorem
Proof Practice
D
4
B
3 5
1
A
Given: AC DB
Prove: m∠1 + m∠2 + m∠3 = 180.
1. AC DB
2. m∠4 + m∠3 + m∠5 = 180.
3. m∠1 = m∠4
4. m∠2 = m∠5
5. m∠1 + m∠2 + m∠3 = 180.
Triangle Angle­Sum Theorem Example
The diagram shows the path
of the softball in a drill developed by four players. Find the measure of each numbered angle.
What relationships do you recognize in the figure?
C
Reasons
Statements
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2
1. Given
2. Straight Angles Equal 180 degrees.
3. Alt. Int. Angles are ≅ 4. Alt. Int. Angles are ≅ 5. Substitution Property
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Triangle Angle­Sum Theorem Example
Step 1) Find m∠1. (Triangle Angle­Sum Thm.) m∠1+ 43 + 74 = 180
m∠1+ 117 = 180
m∠1 = 63
What is a good course of action to solve this problem?
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1
Triangle Angle­Sum Theorem Example
Step 2) Find m∠2. (Vertical Angles are Congruent) Triangle Angle­Sum Theorem Example
Step 3) Find m∠3. (Triangle Angle­Sum Thm.) m∠3 + 63 + 79 = 180
m∠1 = 63
m∠3 + 142 = 180
m∠2 = 63
m∠3 = 38
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Answers
m∠1 = m∠2
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An Exterior Angle is...
Exterior Angle Theorem
An angle formed by one side of a triangle and the extension of another side.
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
Remote Interior Angles are...
The angles of a triangle that are not adjacent to a given exterior angle.
Ex) m∠A + m∠B = m∠1
B
1
A
C
Exterior Angle
Remote Interior
Angles
3
2
1
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Exterior Angle Theorem Proof
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Exterior Angle Theorem Example 1
Find the measure of ∠FLW in the fenced flower garden shown.
What relationships do you recognize in the figure?
What is a good course of action to solve this problem?
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Exterior Angle Theorem Example 1
Exterior Angle Theorem Example 2
Set up equation and solve
for x.
(Exterior Angle Theorem)
The piece of quilt fabric is in the shape of a right triangle. Find the measure of ∠ACD.
x + 32
Answers
m∠LOW + m∠OWL = m∠FLW
= 2x – 48
32 = x – 48
Answers
80 = x
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Triangle Angle­Sum Theorem Corollaries
Triangle Angle Example 1
Corollary­ A statement that can be easily proved using a theorem is called a corollary of that theorem.
1) The acute angles of a right triangle A
are complementary
ex) m∠A + m∠B = 90
C
B
2) There can be at most one right or obtuse angle in a triangle.
Find the measure of each numbered angle.
What relationships do you recognize in the figure?
What is a good course of action to solve this problem?
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Triangle Angle Example 1
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Triangle Angle Example 1
Step 1) Find m∠1.
(Exterior Angle Thm)
Step 3) Find m∠3.
m∠1 = 48 + 56
m∠ 3 = 90 ­ 48 = 104
(Complementary Angles)
= 42
Step 2) Find m∠2. (Linear Pair)
Step 4) Find m∠4. (Triangle Angle­Sum Thm)
m∠1 + m∠2 = 180
(90 – 34) + m∠2 + m∠ 4 = 180
104 + m∠2 = 180
56 + 76 + m∠ 4 = 180
m∠2 = 76
132 + m∠4 = 180
m∠4 = 48
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Triangle Angle Example 1
Triangle Angle Example 2
Step 5) Find m∠5.
(Triangle Angle Sum Thm)
Find m∠3.
m∠5 + 41 + 90 = 180
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Answers
Answers
m∠5 + 131 = 180
m∠5 = 49
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