Download Triangle Sum Theorem

Triangles
Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes
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
Triangles
Problem of the Day
What is the one hundred fiftieth day of
a non-leap year?
May 30
Triangles
Learn to find unknown angles and
identify possible side lengths in
triangles.
Triangles
Vocabulary
Triangle Sum Theorem
acute triangle
right triangle
obtuse triangle
equilateral triangle
isosceles triangle
scalene triangle
Triangle Inequality Theorem
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.
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°.
Triangles
An acute triangle has 3 acute angles. A
right triangle has 1 right angle. An obtuse
triangle has 1 obtuse angle.
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°
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°
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°
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°
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°
Triangles
Check It Out: Example 1C
Find c° in the obtuse triangle.
24° + 38° + c° = 180°
62° + c° = 180°
–62°
–62°
c° = 118°
38°
24°
c°
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.
Triangles
Additional Example 2A: 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°.
Triangles
Additional Example 2B: 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°.
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°.
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°.
t°
t°
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°.
10x°
7x°
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°
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.
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°
Triangle Sum Theorem
Combine like terms.
Divide both sides by 10.
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°
The angles measure 18°,
54°, and 108°. The triangle
is an obtuse scalene
triangle.
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° =
second angle measure, and 1 (3x°) = x° =
3
third angle measures.
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°
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°
36°
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.
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.
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.
Triangles
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
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°
Triangles
Lesson Quiz: Part II
3. Find the missing angle measure in an acute
triangle with angle measures of 67° and 63°.
50°
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
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°
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°
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°