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Angles of Triangles
Objectives
Find angle measures in triangles.
Measures of Angles of a TriangleThe 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 TriangleExtend sides
InteriorAngle
ExteriorAngle
ExteriorAngle
Triangle Interior and Exterior Angles
A
B
C
Smiley faces are interior angles and hearts represent the exterior angles
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
BC
(
)) ((
D
E F
Interior Angles
Exterior Angles(formed by extending the sides)
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.
Draw a triangle and extend one side. Then draw a line parallel to the extended side, as shown.
The three angles in the triangle can be arranged to form a straight line or 180°.
Two sides of the triangle are transversals to the parallel lines.
Triangle Sum Theorem
Theorem 4.1 – Triangle Sum Theorem
The sum of the measures of the angles of a triangle is 180°.
mX + mY + mZ = 180°
X
Y Z
Triangle Sum Theorem
Given mA = 43° and mB = 85°, find mC.
ANSWER C has a measure of 52°.
CHECK Check your solution by substituting 52° for mC. 43° + 85° + 52° = 180°
SOLUTION
mA + mB + mC = 180° Triangle Sum Theorem
43° + 85° + mC = 180° Substitute 43° for mA and 85° for mB.
128° + mC = 180° Simplify.
mC = 52° Simplify.
128° + mC – 128° = 180° – 128° Subtract 128° from each side.
Example 1
A. Find p in the acute triangle.
73° + 44° + p° = 180°
117 + p = 180
p = 63
–117 –117
Triangle Sum Theorem
Subtract 117 from both sides.
Example 2a
B. Find m in the obtuse triangle.
23° + 62° + m° = 180°
85 + m = 180
m = 95
–85 –85
Triangle Sum Theorem
Subtract 85 from both sides.
23
62
m
Example 2b
A. Find a in the acute triangle.
88° + 38° + a° = 180°
126 + a = 180
a = 54
–126 –126
88°
38°
a°
Triangle Sum Theorem
Subtract 126 from both sides.
Your Turn:
B. Find c in the obtuse triangle.
24° + 38° + c° = 180°
62 + c = 180
c = 118
–62 –62 c°
24°
38°Triangle Sum Theorem.
Subtract 62 from both sides.
Your Turn:
2x° + 3x° + 5x° = 180°
10x = 180
x = 18
10 10
Find the angle measures in the scalene triangle.
Triangle Sum Theorem
Simplify.
Divide both sides by 10.
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°.
Example 3
3x° + 7x° + 10x° = 180°
20x = 180
x = 9
20 20
Find the angle measures in the scalene triangle.
Triangle Sum Theorem
Simplify.
Divide both sides by 20.
3x° 7x°
10x°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°.
Your Turn:
ANSWER 65°
ANSWER 75°
ANSWER 50°
Find mA.1.
Find mB.2.
Find mC.3.
Your Turn:
Substitution
Subtract 20 from each side.
Answer:
GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20.
Example 6:
Answer:
The piece of quilt fabric is in the shape of a right triangle. Find if is 62.
Your Turn:
Investigating Exterior Angles of a Triangles
B
A
AB
C
You can put the two torn angles together to exactly cover one of the exterior angles
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
12
34
ANSWER 1 has a measure of 130°.
SOLUTION
m1 = mA + mC Exterior Angle Theorem
Given mA = 58° and mC = 72°, find m1.
Substitute 58° for mA and 72° for mC.
= 58° + 72°
Simplify.= 130°
Example 7
ANSWER 120°
ANSWER 155°
ANSWER 113°
Find m2.1.
Find m3.2.
Find m4.3.
Your Turn:
