Download 7-3 Angles in Triangles

7-3
Angles in Triangles
Warm Up
Problem of the Day
Lesson Presentation
Course 3
7-3 Angles in Triangles
Warm Up
Solve each equation.
1. 62 + x + 37 = 180 x = 81
2. x + 90 + 11 = 180
x = 79
3. 2x + 18 = 180
x = 81
4. 180 = 2x + 72 + x
x = 36
Course 3
7-3 Angles in Triangles
Problem of the Day
What is the one hundred fiftieth day of
a non-leap year?
May 30
Course 3
7-3 Angles in Triangles
Learn to find unknown angles in
triangles.
Course 3
7-3 Insert
Title Here
AnglesLesson
in Triangles
Vocabulary
Triangle Sum Theorem
acute triangle
right triangle
obtuse triangle
equilateral triangle
isosceles triangle
scalene triangle
Course 3
7-3 Angles in Triangles
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.
Course 3
7-3 Angles in Triangles
Draw a triangle and extend one side.
Then draw a line parallel to the
extended side, as shown.
The 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°.
Course 3
7-3 Angles in Triangles
An acute triangle has 3 acute angles. A
right triangle has 1 right angle. An obtuse
triangle has 1 obtuse angle.
Course 3
7-3 Angles in Triangles
Additional Example 1A: Finding Angles in Acute,
Right and Obtuse Triangles
Find p° in the acute triangle.
73° + 44° + p° = 180°
117° + p° = 180°
–117°
–117°
p° = 63°
Course 3
7-3 Angles in Triangles
Additional Example 1B: Finding Angles in Acute,
Right, and Obtuse Triangles
Find c° in the right triangle.
42° + 90° + c° = 180°
132° + c° = 180°
–132°
–132°
c° = 48°
Course 3
7-3 Angles in Triangles
Additional Example 1C: Finding Angles in Acute,
Right, and Obtuse Triangles
Find m° in the obtuse triangle.
23° + 62° + m° = 180°
85° + m° = 180°
–85°
–85°
m° = 95°
Course 3
7-3 Angles in Triangles
Check It Out: Example 1A
Find a° in the acute triangle.
88° + 38° + a° = 180°
38°
126° + a° = 180°
–126°
–126°
a° = 54°
Course 3
a°
88°
7-3 Angles in Triangles
Check It Out: Example 1B
Find b in the right triangle.
38°
38° + 90° + b° = 180°
128° + b° = 180°
–128°
–128°
b° = 52°
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b°
7-3 Angles in Triangles
Check It Out: Example 1C
Find c° in the obtuse triangle.
24° + 38° + c° = 180°
62° + c° = 180°
–62°
–62°
c° = 118°
Course 3
38°
24°
c°
7-3 Angles in Triangles
An equilateral triangle has 3
congruent sides and 3 congruent
angles. An isosceles triangle has at
least 2 congruent sides and 2 congruent
angles. A scalene triangle has no
congruent sides and no congruent
angles.
Course 3
7-3 Angles in Triangles
Additional Example 2A: Finding Angles in Equilateral,
Isosceles, and Scalene Triangles
Find the angle measures in the equilateral
triangle.
3b° = 180° Triangle Sum Theorem
3b° 180°
=
3
3
Divide both
sides by 3.
b° = 60°
All three angles measure 60°.
Course 3
7-3 Angles in Triangles
Additional Example 2B: Finding Angles in Equilateral,
Isosceles, and Scalene Triangles
Find the angle measures in the isosceles
triangle.
62° + t° + t° = 180°
62° + 2t° = 180°
–62°
–62°
Triangle Sum Theorem
Combine like terms.
Subtract 62° from both sides.
2t° = 118°
2t° = 118°
Divide both sides by 2.
2
2
t° = 59°
The angles labeled t° measure 59°.
Course 3
7-3 Angles in Triangles
Additional Example 2C: Finding Angles in Equilateral,
Isosceles, and Scalene Triangles
Find the angle measures in the scalene
triangle.
2x° + 3x° + 5x° = 180°
10x° = 180°
10
10
Triangle Sum Theorem
Combine like terms.
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°.
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7-3 Angles in Triangles
Check It Out: Example 2A
Find the angle measures in the isosceles
triangle.
39° + t° + t° = 180° Triangle Sum Theorem
39° + 2t° = 180° Combine like terms.
–39°
–39° Subtract 39° from both sides.
2t° = 141°
2t° = 141°
Divide both sides by 2
2
2
39°
t° = 70.5°
The angles labeled t° measure 70.5°.
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t°
t°
7-3 Angles in Triangles
Check It Out: Example 2B
Find the angle measures in the scalene
triangle.
3x° + 7x° + 10x° = 180° Triangle Sum Theorem
20x° = 180° Combine like terms.
20
20 Divide both sides by 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
3x°
10(9°) = 90°.
Course 3
10x°
7x°
7-3 Angles in Triangles
Check It Out: Example 2C
Find the angle measures in the equilateral
triangle.
3x° = 180°
Triangle Sum Theorem
3x° 180°
=
3
3
x°
x° = 60°
All three angles measure 60°.
Course 3
x°
x°
7-3 Angles in Triangles
Additional Example 3: Finding Angles in a Triangle
that Meets Given Conditions
The second angle in a triangle is six times
as large as the first. The third angle is half
as large as the second. Find the angle
measures and draw a possible picture.
Let x° = the first angle measure. Then 6x° =
second angle measure, and 1 (6x°) = 3x° =
2
third angle measure.
Course 3
7-3 Angles in Triangles
Additional Example 3 Continued
Let x° = the first angle measure. Then 6x° =
second angle measure, and 1 (6x°) = 3x° =
2
third angle.
x° + 6x° + 3x° = 180°
10x° = 180°
10
10
x° = 18°
Course 3
Triangle Sum Theorem
Combine like terms.
Divide both sides by 10.
7-3 Angles in Triangles
Additional Example 3 Continued
Let x° = the first angle measure. Then 6x° =
second angle measure, and 1 (6x°) = 3x° =
2
third angle.
x° = 18°
3 • 18° = 54°
6 • 18° = 108°
X° = 18°
Course 3
The angles measure 18°,
54°, and 108°. The triangle
is an obtuse scalene
triangle.
7-3 Angles in Triangles
Check It Out: Example 3
The second angle in a triangle is three
times larger than the first. The third
angle is one third as large as the second.
Find the angle measures and draw a
possible picture.
Let x° = the first angle measure. Then 3x° =
second angle measure, and 1 (3x°) = x° =
3
third angle measures.
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7-3 Angles in Triangles
Check It Out: Example 3 Continued
Let x° = the first angle measure. Then 3x° =
second angle measure, and 1 (3x°) = 3x° =
3
third angle.
x° + 3x° + x° = 180°
Triangle Sum Theorem
5x° = 180°
5
5
Combine like terms.
Divide both sides by 5.
x° = 36°
Course 3
7-3 Angles in Triangles
Check It Out: Example 3 Continued
Let x° = the first angle measure. Then 3x° =
second angle measure, and 1 (3x°) = x° =
3
third angle.
x° = 36°
3 • 36° = 108°
x° = 36°
The angles measure 36°,
36°, and 108°. The triangle
is an obtuse isosceles
triangle.
108°
36°
Course 3
36°
7-3 Angles in Triangles
Lesson Quiz: Part I
1. Find the missing angle measure in the
acute triangle shown. 38°
2. Find the missing angle measure in the
right triangle shown. 55°
Course 3
7-3 Angles in Triangles
Lesson Quiz: Part II
3. Find the missing angle measure in an acute
triangle with angle measures of 67° and 63°.
50°
4. Find the missing angle measure in an obtuse
triangle with angle measures of 10° and 15°.
155°
Course 3