1. 5-3 Triangles Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of the Day Problem of the Day Lesson Quizzes Lesson Quizzes

2. 5-3 Triangles Warm Up Solve each equation. 1. 62 + x + 37 = 180 2. x + 90 + 11 = 180 3. 2x + 18 = 180 4. 180 = 2x + 72 + x x = 81 x = 79 x = 81 x = 36

3. 5-3 Triangles Problem of the Day What is the one hundred fiftieth day of a non-leap year? May 30

4. 5-3 Triangles Learn to find unknown angles and identify possible side lengths in triangles.

5. 5-3 Triangles Vocabulary Triangle Sum Theorem acute triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle Triangle Inequality Theorem

6. 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.

7. 5-3 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.

8. 5-3 Triangles An acute triangle has 3 acute angles. A right triangle has 1 right angle. An obtuse triangle has 1 obtuse angle.

9. 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° c° = 48° –132°

10. 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° m° = 95° –85° –85°

11. 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° p° = 63° –117°

12. 5-3 Triangles Find b in the right triangle. 38° + 90° + b° = 180° 128° + b° = 180° b° = 52° –128° 38° b°b° Check It Out: Example 1A

13. 5-3 Triangles Check It Out: Example 1B Find a° in the acute triangle. 88° + 38° + a° = 180° 126° + a° = 180° a° = 54° –126° 88° 38° a°a°

14. 5-3 Triangles Find c° in the obtuse triangle. 24° + 38° + c° = 180° 62° + c° = 180° c° = 118° –62° –62° c°c° 24° 38° Check It Out: Example 1C

15. 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.

16. 5-3 Triangles Additional Example 2A: 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 t° = 59° Triangle Sum Theorem Combine like terms. Subtract 62° from both sides. Divide both sides by 2. The angles labeled t° measure 59°.

17. 5-3 Triangles Additional Example 2B: 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°.

18. 5-3 Triangles Additional Example 2C: Finding Angles in Equilateral, Isosceles, and Scalene Triangles Find the angle measures in the equilateral triangle. 3b° = 180° b° = 60° 3b° 180° 3 = Triangle Sum Theorem All three angles measure 60°. Divide both sides by 3.

19. 5-3 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 t° = 70.5° Triangle Sum Theorem Combine like terms. Subtract 39° from both sides. Divide both sides by 2 t° 39° The angles labeled t° measure 70.5°.

20. 5-3 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°3x°7x°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°.

21. 5-3 Triangles Find the angle measures in the equilateral triangle. 3x° = 180° x° = 60° 3x° 180° 3 = Triangle Sum Theorem All three angles measure 60°. Check It Out: Example 2C x°x° x°x° x°x°

22. 5-3 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. 1212 Additional Example 3: Finding Angles in a Triangle that Meets Given Conditions

23. 5-3 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. 1212

24. 5-3 Triangles 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. 1212

25. 5-3 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 figure. Check It Out: Example 3 Let x° = the first angle measure. Then 3x° = second angle measure, and (3x°) = x° = third angle measures. 1313

26. 5-3 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. 1313

27. 5-3 Triangles x° = 36° 3 36° = 108° The angles measure 36°, 36°, and 108°. The triangle is an obtuse isosceles triangle. 36° 108° Check It Out: Example 3 Continued Let x° = the first angle measure. Then 3x° = second angle measure, and (3x°) = x° = third angle. 1313

28. 5-3 Triangles Tell whether a triangle can have sides with the given lengths. Explain. Find the sum of the lengths of each pair of sides and compare it to the third side. Additional Example 4A: Using the Triangle Inequality Theorem 8 ft, 10 ft, 13 ft 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.

29. 5-3 Triangles Tell whether a triangle can have sides with the given lengths. Explain. Find the sum of the lengths of each pair of sides and compare it to the third side. Additional Example 4B: Using the Triangle Inequality Theorem 2 m, 4 m, 6 m 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.

30. 5-3 Triangles Tell whether a triangle can have sides with the given lengths. Explain. Find the sum of the lengths of each pair of sides and compare it to the third side. Check It Out: Example 4 17 m, 15 m, 33 m 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.

31. 5-3 Triangles Standard Lesson Quiz Lesson Quizzes Lesson Quiz for Student Response Systems

32. 5-3 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°

33. 5-3 Triangles Lesson Quiz: Part II 3. Find the missing angle measure in an acute triangle with angle measures of 67° and 63°. 4. Tell whether a triangle can have sides with lengths of 4 cm, 8 cm, and 12 cm. 50° No; 4 + 8 is not greater than 12

34. 5-3 Triangles 1. Identify the missing angle measure in the acute triangle shown. A. 43° B. 57° C. 80° D. 90° Lesson Quiz for Student Response Systems

35. 5-3 Triangles 2. Identify the missing angle measure in the acute triangle shown. A. 40° B. 50° C. 90° D. 180° Lesson Quiz for Student Response Systems

36. 5-3 Triangles 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° Lesson Quiz for Student Response Systems