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3.4 Triangle Angle Theorems
I CAN...Apply the Triangle AngleSum Theorem and Exterior Angle Theorem
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Triangle AngleSum Theorem
The sum of the measures of the angles of a triangle is 180.
Ex) m∠A + m∠B + m∠C = 180.
A
B
C
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Triangle AngleSum Theorem Proof
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1. AC DB2. m∠4 + m∠3 + m∠5 = 180.3. m∠1 = m∠44. m∠2 = m∠55. m∠1 + m∠2 + m∠3 = 180.
Statements Reasons1. Given2. Straight Angles Equal 180 degrees.3. Alt. Int. Angles are ≅ 4. Alt. Int. Angles are ≅ 5. Substitution Property
Triangle AngleSum TheoremProof Practice
A
B
C
D54 3
21Given: AC DBProve: m∠1 + m∠2 + m∠3 = 180.
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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?
What is a good course of action to solve this problem?
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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
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Triangle AngleSum Theorem Example
Step 2) Find m∠2. (Vertical Angles are Congruent)
m∠1 = m∠2
m∠1 = 63
m∠2 = 63
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Triangle AngleSum Theorem Example
Step 3) Find m∠3. (Triangle AngleSum Thm.)
m∠3 + 63 + 79 = 180
m∠3 + 142 = 180
m∠3 = 38
Answers
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An Exterior Angle is...
An angle formed by one side of a triangle and the extension of another side.
Remote Interior Angles are...
The angles of a triangle that are not adjacent to a given exterior angle.
12
3Exterior Angle
Remote InteriorAngles
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Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior
angles.
Ex) m∠A + m∠B = m∠11
A
B
C
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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?
3
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Exterior Angle Theorem Example 1Set up equation and solvefor x.(Exterior Angle Theorem)
m∠LOW + m∠OWL = m∠FLW
x + 32 = 2x – 48
32= x – 48
80= x
Answers
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Exterior Angle Theorem Example 2
The piece of quilt fabric is in the shape of a right triangle. Find the measure of ∠ACD.
Answers
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Triangle AngleSum Theorem Corollaries
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 are complementaryex) m∠A + m∠B = 90
2) There can be at most one right or obtuse angle in a triangle.
A
C B
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Triangle Angle Example 1
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
Step 1) Find m∠1.(Exterior Angle Thm)
m∠1 = 48 + 56
= 104
Step 2) Find m∠2. (Linear Pair)
m∠1 + m∠2 = 180
104 + m∠2 = 180
m∠2 = 76
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Triangle Angle Example 1
Step 3) Find m∠3.(Complementary Angles)
m∠ 3 = 90 48
= 42
Step 4) Find m∠4. (Triangle AngleSum Thm)
(90 – 34) + m∠2 + m∠ 4 = 180
56 + 76 + m∠ 4 = 180
132 + m∠4 = 180
m∠4 = 48