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Transcript
Angles of Triangles
Section 4.2
Objectives
Find angle measures in triangles.
Key Vocabulary
Corollary
Exterior angles
Interior angles
Theorems
4.1 Triangle Sum Theorem

Corollary to the Triangle Sum Theorem
4.2 Exterior Angle Theorem
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
Each vertex has a pair
of congruent exterior
angles; however it is
common to show only
one exterior angle at
each vertex.
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.
Triangle Sum Theorem
If you tear off two corners of a triangle and
place them next to the third corner, the
three angles seem to form a straight line.
You can also show this in a drawing.
Triangle Sum Theorem
Draw a triangle and extend one side. Then
draw a line parallel to the extended side, as
shown.
Two sides of the
triangle are
transversals to the
parallel lines.
The three angles in the triangle can be
arranged to form a straight line or 180°.
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°
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
–117
–117
p = 63
Triangle Sum
Theorem
Subtract 117 from
both sides.
Example 2b
B. Find m in the obtuse triangle.
62
23° + 62° + m° = 180°
Triangle Sum
Theorem
23
85 + m = 180
–85
–85
m = 95
Subtract 85 from
both sides.
m
A. Find a in the acute triangle.
88° + 38° + a° = 180°
126 + a = 180
–126
–126
a = 54
Triangle Sum
Theorem
38°
Subtract 126
from both sides.
a°
88°
B. Find c in the obtuse triangle.
24° + 38° + c° = 180°
62 + c = 180
–62
–62
c = 118
Triangle Sum
Theorem.
38°
24°
Subtract 62 from
both sides.
c°
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°.
Find the angle measures in the scalene triangle.
3x° + 7x° + 10x° = 180°
20x = 180
20
20
x=9
The angle labeled 3x°
measures 3(9°) = 27°, the
angle labeled 7x°
measures 7(9°) = 63°, and
the angle labeled 10x°
measures 10(9°) = 90°.
Triangle Sum Theorem
Simplify.
Divide both sides by 20.
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.
Find the missing angle measures.
Corollaries
Definition: A corollary is a theorem with a
proof that follows as a direct result of
another theorem.
As a theorem, a corollary can be used as
a reason in a proof.
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.
1. Find mA.
65°
75°
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.
The piece of quilt fabric is in the shape of a
right triangle. Find
if
is 62.
Exterior Angles and Triangles
An exterior angle is formed by one side of a
triangle and the extension of another side
(i.e. 1 ).
2
1
4
3
The interior angles of the triangle not adjacent to
a given exterior angle are called the remote
interior angles (i.e. 2 and 3).
Investigating Exterior Angles of a
Triangles
You can put the two torn angles
together to exactly cover one of the
exterior angles
B
A
B
C
A
Theorem 4.2 – 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.
m 1 = m 2 + m 3
2
1
4
3
Example 7
Given mA = 58° and mC = 72°, find m1.
SOLUTION
m1 = mA + mC
Exterior Angle Theorem
= 58° + 72°
Substitute 58° for mA and
72° for mC.
= 130°
Simplify.
1 has a measure of 130°.
1. Find m2.
120°
155°
113°
2. Find m3.
3. Find m4.
Example 8:
Find the measure of each numbered angle in the figure.
Exterior Angle Theorem
Simplify.
If 2 s form a linear pair, they
are supplementary.
Substitution
Subtract 70 from each side.
Example 8:
m∠1=70
m∠2=110
Exterior Angle Theorem
Substitution
Subtract 64 from each side.
If 2 s form a linear pair,
they are supplementary.
Substitution
Simplify.
Subtract 78 from each side.
m∠1=70
m∠2=110
m∠3=46
m∠4=102
Example 8:
Angle Sum Theorem
Substitution
Simplify.
Subtract 143 from each side.