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Angles of Triangles
Section 4.2
Objectives
Find angle measures in triangles.
Key Vocabulary
Corollary
Exterior angles
Interior angles
Measures of Angles of a Triangle
The word “triangle” means “three angles”




When the sides of a triangles are extended,
however, other angles are formed
The original 3 angles of the triangle are the
interior angles
The angles that are adjacent to interior angles
are the exterior angles
Each vertex has a pair of exterior angles
Original Triangle
Extend sides
Exterior
Angle
Exterior
Angle
Interior
Angle
Triangle Interior and Exterior Angles
Smiley faces are interior
angles and hearts
represent the exterior
angles
B
A
C
Triangle Interior and Exterior Angles
A
)))

Interior Angles
C
B
(

D
Exterior Angles
(formed by extending the sides)
E
F
Triangle Sum Theorem
The Triangle Angle-Sum Theorem gives
the relationship among the interior angle
measures of any triangle.
Theorem 4.1 – Triangle Sum Theorem
The sum of the measures of the angles of a
triangle is 180°.
X
mX + mY + mZ = 180°
Y
Z
Triangle Sum Theorem
Example 1
Given mA = 43° and mB = 85°, find mC.
SOLUTION
mA + mB + mC = 180°
43° + 85° + mC = 180°
128° + mC = 180°
128° + mC – 128° = 180° – 128°
mC = 52°
ANSWER
CHECK
Triangle Sum Theorem
Substitute 43° for mA and
85° for mB.
Simplify.
Subtract 128° from each side.
Simplify.
C has a measure of 52°.
Check your solution by substituting 52° for mC. 43° +
85° + 52° = 180°
Example 2a
A. Find p in the acute triangle.
73° + 44° + p° = 180°
117 + p = 180
Triangle Sum
Theorem
Example 2b
B. Find m in the obtuse triangle.
62
23° + 62° + m° = 180°
Triangle Sum
Theorem
23
m
Your Turn:
A. Find a in the acute triangle.
88° + 38° + a° = 180°
Triangle Sum
Theorem
38°
a°
88°
Your Turn:
B. Find c in the obtuse triangle.
24° + 38° + c° = 180°
Triangle Sum
Theorem.
38°
24°
c°
Properties of Triangles
Now do these:
41o
80o
30o
b=
c
c=
79o
y z
x
34o
62o
54o
a
a=
141o
b
q
57o
58o
x = 180 – 141 = 39
y = 180 – (58+39)
= 83
z = 180 – 83 = 97
p
p = 180 – (90+57) = 33
r
q = 57
(vertically opposite
angles are equal)
r = 180 – (79+57) = 44
Example 3
Find the angle measures in the scalene triangle.
2x° + 3x° + 5x° = 180°
10x = 180
10
10
Triangle Sum Theorem
Simplify.
Divide both sides by 10.
x = 18
The angle labeled 2x° measures
2(18°) = 36°, the angle labeled
3x° measures 3(18°) = 54°, and
the angle labeled 5x° measures
5(18°) = 90°.
Your Turn:
Find the angle measures in the scalene triangle.
3x° + 7x° + 10x° = 180°
Triangle Sum Theorem
The angle labeled 3x°
measures _____ = __°,
the angle labeled 7x°
measures ____= ___°,
and the angle labeled 10x°
measures _____ = ___°.
10x°
3x°
7x°
Example 4:
Find the missing angle measures.
Find
first because the
measure of two angles of
the triangle are known.
Angle Sum Theorem
Simplify.
Subtract 117 from each side.
Example 4:
Angle Sum Theorem
Simplify.
Subtract 142 from each side.
Answer:
Your Turn:
Find the missing angle measures.
Answer:
Triangle Angle-Sum Corollaries
Corollary 4.1 – The acute s of a right ∆
are complementary.
Example: m∠x + m∠y = 90˚
x°
y°
Example 5
∆ABC and ∆ABD are right triangles.
Suppose mABD = 35°.
a. Find mDAB.
b. Find mBCD.
SOLUTION
a. mDAB + mABD = 90°
mDAB + 35° = 90°
mDAB + 35° – 35° = 90° – 35°
mDAB = 55°
b. mDAB + mBCD = 90°
55° + mBCD = 90°
mBCD = 35°
Corollary to the
Triangle Sum Theorem
Substitute 35° for mABD.
Subtract 35° from each side.
Simplify.
Corollary to the
Triangle Sum Theorem
Substitute 55° for mDAB.
Subtract 55° from each side.
Your Turn:
1. Find mA.
ANSWER
65°
ANSWER
75°
ANSWER
50°
2. Find mB.
3.
Find mC.
Example 6:
GARDENING The flower bed shown is in the shape of
a right triangle. Find
if
is 20.
Corollary 4.1
Substitution
Subtract 20 from each side.
Answer:
Your Turn:
The piece of quilt fabric is in the shape of a
right triangle. Find
if
is 62.
Answer:
Joke Time
What's orange and sounds like a parrot?
A carrot!
What do you call cheese that doesn't belong to
you?
Nacho cheese.
Why do farts smell?
So the deaf can enjoy them too.
Assignment
IXL W6
IXL W11
IXL W 12
IXL W 15
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