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7-3 Angles in Triangles
Course 3
Warm Up
Problem of the Day
Lesson Presentation
Warm UpSolve each equation.1. 62 + x + 37 = 180
2. x + 90 + 11 = 180
3. 2x + 18 = 180
4. 180 = 2x + 72 + x
Course 3
7-3 Angles in Triangles
x = 81
x = 79
x = 81
x = 36
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 Angles in Triangles
VocabularyTriangle Sum Theorem
acute triangle
right triangle
obtuse triangle
equilateral triangle
isosceles triangle
scalene triangle
Insert Lesson Title Here
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 three angles in the triangle can be arranged to form a straight line or 180°.
The sides of the triangle are transversals to the parallel lines.
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°
p° = 63°
–117° –117°
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°
c° = 48°
–132° –132°
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°
m° = 95°
–85° –85°
Course 3
7-3 Angles in Triangles
Check It Out: Example 1A
Find a° in the acute triangle.
88° + 38° + a° = 180°
126° + a° = 180°
a° = 54°
–126° –126°
88°
38°
a°
Course 3
7-3 Angles in Triangles
Find b in the right triangle.
38° + 90° + b° = 180°
128° + b° = 180°
b° = 52°
–128° –128°
38°
b°
Check It Out: Example 1B
Course 3
7-3 Angles in Triangles
Find c° in the obtuse triangle.
24° + 38° + c° = 180°
62° + c° = 180°
c° = 118°
–62° –62° c°24°
38°
Check It Out: Example 1C
Course 3
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°
b° = 60°
3b° 180°3 3
=
Triangle Sum Theorem
All three angles measure 60°.
Divide both sides by 3.
Course 3
7-3 Angles in Triangles
Additional Example 2B: Finding Angles in Equilateral, Isosceles, and Scalene Triangles
62° + t° + t° = 180°62° + 2t° = 180°
2t° = 118°
–62° –62°
Find the angle measures in the isosceles triangle.
2t° = 118°2 2
t° = 59°
Triangle Sum TheoremCombine like terms.Subtract 62° from both sides.
Divide both sides by 2.
The angles labeled t° measure 59°.Course 3
7-3 Angles in Triangles
Additional Example 2C: Finding Angles in Equilateral, Isosceles, and Scalene Triangles
2x° + 3x° + 5x° = 180°
10x° = 180°
x = 18°
10 10
Find the angle measures in the scalene triangle.
Triangle Sum Theorem
Combine like terms.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°.
Course 3
7-3 Angles in Triangles
Check It Out: Example 2A
39° + t° + t° = 180°39° + 2t° = 180°
2t° = 141°
–39° –39°
Find the angle measures in the isosceles triangle.
2t° = 141°2 2
t° = 70.5°
Triangle Sum TheoremCombine like terms.Subtract 39° from both sides.
Divide both sides by 2
t°t°
39°
The angles labeled t° measure 70.5°.
Course 3
7-3 Angles in Triangles
3x° + 7x° + 10x° = 180°
20x° = 180°
x = 9°
20 20
Find the angle measures in the scalene triangle.
Triangle Sum Theorem
Combine like terms.Divide both sides by 20.
3x° 7x°
10x°
Check It Out: Example 2B
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°.
Course 3
7-3 Angles in Triangles
Find the angle measures in the equilateral triangle.
3x° = 180°
x° = 60°
3x° 180°3 3
=
Triangle Sum Theorem
All three angles measure 60°.
Check It Out: Example 2C
x° x°
x°
Course 3
7-3 Angles in Triangles
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 (6x°) = 3x° =
third angle measure.
12
Additional Example 3: Finding Angles in a Triangle that Meets Given Conditions
Course 3
7-3 Angles in Triangles
Additional Example 3 Continued
x° + 6x° + 3x° = 180°
10x° = 180° 10 10
x° = 18°
Triangle Sum Theorem
Combine like terms.Divide both sides by 10.
Let x° = the first angle measure. Then 6x° = second angle measure, and (6x°) = 3x° = third angle.
12
Course 3
7-3 Angles in Triangles
X° = 18°
x° = 18°
6 • 18° = 108°3 • 18° = 54°
The angles measure 18°, 54°, and 108°. The triangle is an obtuse scalene triangle.
Additional Example 3 Continued
Let x° = the first angle measure. Then 6x° = second angle measure, and (6x°) = 3x° = third angle.
12
Course 3
7-3 Angles in Triangles
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.
Check It Out: Example 3
Let x° = the first angle measure. Then 3x° =
second angle measure, and (3x°) = x° =
third angle measures.
13
Course 3
7-3 Angles in Triangles
x° + 3x° + x° = 180°
5x° = 180° 5 5
x° = 36°
Triangle Sum Theorem
Combine like terms.Divide both sides by 5.
Check It Out: Example 3 Continued
Let x° = the first angle measure. Then 3x° = second angle measure, and (3x°) = 3x° = third angle.
13
Course 3
7-3 Angles in Triangles
x° = 36°
x° = 36°3 • 36° = 108°
The angles measure 36°, 36°, and 108°. The triangle is an obtuse isosceles triangle.
36° 36°
108°
Check It Out: Example 3 Continued
Let x° = the first angle measure. Then 3x° = second angle measure, and (3x°) = x° = third angle.
13
Course 3
7-3 Angles in Triangles
Lesson Quiz: Part I
1. Find the missing angle measure in the acute triangle shown.
2. Find the missing angle measure in the right triangle shown.
38°
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°.
4. Find the missing angle measure in an obtuse triangle with angle measures of 10° and 15°.
50°
155°
Course 3
7-3 Angles in Triangles