1. Learn to find unknown angles and identify possible side lengths in triangles.

2. Vocabulary Triangle Sum Theorem acute triangle right triangle
obtuse triangle
equilateral triangle
isosceles triangle
scalene triangle
Triangle Inequality Theorem

3. 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.

4. An acute triangle has 3 acute angles
An acute triangle has 3 acute angles. A right triangle has 1 right angle. An obtuse triangle has 1 obtuse angle.

5. 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°

6. 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°

7. 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°

8. Check It Out: Example 1A
Find b in the right triangle.
38°
38° + 90° + b° = 180°
128° + b° = 180°
–128° –128°
b° = 52° b°

9. An equilateral triangle has 3 congruent sides and 3 congruent angles
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.

10. 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
62° + 2t° = 180°
Combine like terms.
–62° –62°
Subtract 62° from both sides.
2t° = 118°
2t° = 118°
Divide both sides by 2.
t° = 59°
The angles labeled t° measure 59°.

11. Find the angle measures in the scalene triangle.
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.
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°.

12. 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° ° =
Divide both sides by 3.
b° = 60°
All three angles measure 60°.

13. 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
39°
t° = 70.5° t°
The angles labeled t° measure 70.5°. t°

14. Find the angle measures in the scalene triangle.
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.
Divide both sides by 20.
x = 9°
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°.
3x°
7x°

15. Check It Out: Example 2C
Find the angle measures in the equilateral triangle.
3x° = 180°
Triangle Sum Theorem
3x° ° = x°
x° = 60° x° x°
All three angles measure 60°.

16. 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. ? ? ?
> 13
> 13
> 10
18 > 13 
23 > 13 
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.

17. 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.

18. 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. ?
> 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.

19. 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°

20. 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; is not greater than 12