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5-3 Triangles
Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes
5-3 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
5-3 Triangles
Problem of the Day
What is the one hundred fiftieth day of
a non-leap year?
May 30
5-3 Triangles
Learn to find unknown angles and
identify possible side lengths in
triangles.
5-3 Triangles
Vocabulary
Triangle Sum Theorem
acute triangle
right triangle
obtuse triangle
equilateral triangle
isosceles triangle
scalene triangle
Triangle Inequality Theorem
5-3 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.
5-3 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°.
5-3 Triangles
An acute triangle has 3 acute angles. A
right triangle has 1 right angle. An obtuse
triangle has 1 obtuse angle.
5-3 Triangles
Additional Example 1A: 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°
5-3 Triangles
Additional Example 1B: 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°
5-3 Triangles
Additional Example 1C: 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°
5-3 Triangles
Check It Out: Example 1A
Find b in the right triangle.
38°
38° + 90° + b° = 180°
128° + b° =
180°
–128°
–128°
b° = 52°
b
°
5-3 Triangles
Check It Out: Example 1B
Find a° in the acute triangle.
88° + 38° + a° = 180°
38°
126° + a° =
180°
–126°
–126°
a° = 54°
a
°
88°
5-3 Triangles
Check It Out: Example 1C
Find c° in the obtuse triangle.
24° + 38° + c° = 180°
62° + c° =
180°
–62°
–62°
c° = 118°
24
°
38
°
c
°
5-3 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.
5-3 Triangles
Additional Example 2A: Finding Angles in Equilateral,
Isosceles, and Scalene Triangles
Find the angle measures in the isosceles
triangle.
62° + t° + t° = 180°
Triangle Sum Theorem
Combine like terms.
62° + 2t° =
180°
–62°
–62° Subtract 62° from both
2t° = 118° sides.
2t° = 118°
Divide both sides by 2.
2
2
t° = 59°
The angles labeled t° measure 59°.
5-3 Triangles
Additional Example 2B: Finding Angles in Equilateral,
Isosceles, and Scalene Triangles
Find the angle measures in the scalene
triangle.
2x° + 3x° + 5x° = 180°
Triangle Sum Theorem
10x° = 180° Combine like terms.
10
10
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°.
5-3 Triangles
Additional Example 2C: 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°.
5-3 Triangles
Check It Out: Example 2A
Find the angle measures in the isosceles
triangle.
39° + t° + t° = 180°
Triangle Sum Theorem
Combine like terms.
39° + 2t° =
180°
–39°
–39° Subtract 39° from both
2t° = 141° sides.
2t° = 141°
Divide both sides by 2
2
2
t° =
70.5°
The angles labeled t° measure 70.5°. t°
39°
t°
5-3 Triangles
Check It Out: Example 2B
Find the angle measures in the scalene
triangle.
3x° + 7x° + 10x° =
Triangle Sum Theorem
180°
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°.
°
10x°
7x°
5-3 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°.
x°
x°
5-3 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°
1
= second angle measure, and
(6x°) =
2
3x° = third angle measure.
5-3 Triangles
Additional Example 3 Continued
Let x° = the first angle measure. Then 6x°
1
= second angle measure, and
(6x°) =
2
3x° = third angle.
x° + 6x° + 3x° = 180° Triangle Sum Theorem
10x° = 180° Combine like terms.
10
10
Divide both sides by 10.
x° = 18°
5-3 Triangles
Additional Example 3 Continued
Let x° = the first angle measure. Then 6x°
1
= second angle measure, and
(6x°) =
2
3x° = third angle.
x° = 18° The angles measure 18°,
3 • 18° = 54° 54°, and 108°. The
6 • 18° = 108° triangle is an obtuse
scalene triangle.
5-3 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 figure.
Let x° = the first angle measure. Then 3x°
1
= second angle measure, and
(3x°) = x°
3
= third angle measures.
5-3 Triangles
Check It Out: Example 3 Continued
Let x° = the first angle measure. Then 3x°
1
= second angle measure, and
(3x°) =
3
3x° = third angle.
x° + 3x° + x° =
180°
5x° = 180°
5
5
x° = 36°
Triangle Sum Theorem
Combine like terms.
Divide both sides by 5.
5-3 Triangles
Check It Out: Example 3 Continued
Let x° = the first angle measure. Then 3x°
1
= second angle measure, and
(3x°) =
3
x° = third angle.
The angles measure 36°,
x° = 36°
36°, and 108°. The
3 • 36° = 108° triangle is an obtuse
x° = 36° isosceles triangle.
108°
36°
36°
5-3 Triangles
Additional Example 4A: Using the Triangle Inequality
Theorem
Tell whether a triangle can have sides with
the given lengths. Explain.
8 ft, 10 ft, 13 ft
Find the sum of the lengths of each pair of
sides and compare it to the third side.
?
8 + 10 > 13
18 > 13 
?
10 + 13 > 13
23 > 13 
?
8 + 13 > 10
21 > 10 
A triangle can have these side lengths. The
sum of the lengths of any two sides is greater
than the length of the third side.
5-3 Triangles
Additional Example 4B: Using the Triangle Inequality
Theorem
Tell whether a triangle can have sides with
the given lengths. Explain.
2 m, 4 m, 6 m
Find the sum of the lengths of each pair of
sides and compare it to the third side.
?
2+4>6
6>6
A triangle cannot have these side lengths.
The sum of the lengths of two sides is not
greater than the length of the third side.
5-3 Triangles
Check It Out: Example 4
Tell whether a triangle can have sides with
the given lengths. Explain.
17 m, 15 m, 33 m
Find the sum of the lengths of each pair of
sides and compare it to the third side.
?
17 + 15 > 33
32 > 33 
A triangle cannot have these side lengths.
The sum of the lengths of two sides is not
greater than the length of the third side.
5-3 Triangles
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
5-3 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°
5-3 Triangles
Lesson Quiz: Part II
3. Find the missing angle measure in an acute
triangle with angle measures of 67° and
50°
63°.
4. Tell whether a triangle can have sides with
lengths of 4 cm, 8 cm, and 12 cm.
No; 4 + 8 is not greater than 12
5-3 Triangles
Lesson Quiz for Student Response Systems
1. Identify the missing angle measure in the acute
triangle shown.
A. 43°
B. 57°
C. 80°
D. 90°
5-3 Triangles
Lesson Quiz for Student Response Systems
2. Identify the missing angle measure in the acute
triangle shown.
A. 40°
B. 50°
C. 90°
D. 180°
5-3 Triangles
Lesson Quiz for Student Response Systems
3. Identify the missing angle measure in an acute
triangle with angle measures of 38° and 61°.
A. 38°
B. 61°
C. 81°
D. 99°