G8 Math Q4 - Week 1 - Illustrating the triangle inequality.pptx

Illustrating the triangle inequality

1. Illustrating Theorems on Triangle Inequalities

2. Investigate Me!

3. Directions: Use the figure below to answer the questions that follow. 1. What is the included side in ∠ B and ∠ C? in ∠ E and ∠ F? 2. What is the included angle in 𝐴𝐶̅̅̅̅ and 𝐵𝐶̅̅̅̅ ? in 𝐷𝐹̅̅̅̅ and 𝐸𝐹̅̅̅̅ ?      

4. Directions: Use the figure below to answer the questions that follow. 2. What is the included angle in and ? in and ?   3. What is the sum of the interior angles of ΔABC? ΔDEF? 1. What is the included side in ∠ B and ∠ C? in ∠ E and ∠ F?      

5. Directions: Use the figure below to answer the questions that follow. 3. What is the sum of the interior angles of ΔABC? ΔDEF? 4. If ∠B ≅ ∠E, and ∠C ≅ ∠F, what additional information is required to tell that the triangles are congruent using SAS Congruence? 2. What is the included angle in and ? in and ?    

6. Directions: Use the figure below to answer the questions that follow. 4. If ∠B ≅ ∠E, and ∠C ≅ ∠F, what additional information is required to tell that the triangles are congruent using ASA Congruence? 5. If ≅ and ≅ , what additional information is required to tell that the triangles are congruent using SSS Congruence?   3. What is the sum of the interior angles of ΔABC? ΔDEF?  

7. Directions: Use the figure below to answer the questions that follow. 5. If ≅ and ≅ , What additional information is required to tell that the triangles are congruent using SSS Congruence?   4. If ∠B ≅ ∠E, and ∠C ≅ ∠F, what additional information is required to tell that the triangles are congruent using SAS Congruence?      

8. Inequalities in One Triangle

9. The most used theorems in one triangle: Angle-Side Relationship Theorem Triangle Inequality Theorem Exterior Angle Inequality Theorem Hinge Theorem (SAS Inequality Theorem) Converse of Hinge Theorem (SSS Inequality Theorem) Inequalities in One Triangle

10.   If two angles of a triangle are not congruent, then the larger side is opposite the larger angle and vice versa.

11. Example 1: Compare the length of the sides of the following triangle. Step 1: Find the measure of the third angle. The sum of all the angles in any triangle is 180º. ∠𝐹 + ∠𝑈 + ∠𝑁 = 180° 50 O + 105+ ∠𝑁 = 180° ∠𝑁 + 155° = 180° ∠𝑁 = 180°−155° ∠𝑁 = 25°

12. Example 1: Step 2: Look at the relative sizes of the angles and compare. ∠𝑁 < ∠𝐹 < ∠𝑈 Compare the length of the sides of the following triangle. 25° Step 3: Following the angle-side relationship we can order the sides accordingly. Remember it is the side opposite the angle. < <   Thus , is the longest side since it is the opposite side of the largest angle, ∠ U, while 𝐹𝑈̅̅̅̅ is the shortest side whose opposite angle, ∠ N measures 25°.  

13. Example 2: Compare the measure of the angles of the following triangle. < <      

14. Compare the sides of each triangle in ascending order. U I E b. 46  P O N a. 59  61  M E L c. 70  P A T d. 79  42  J R E e. 31  73            50 

15.   The sum of the lengths of any two sides of a triangle is greater than the length of the remaining side.

16.   Let a, b, and c be the lengths of a triangle. These lengths may only form a triangle if the three conditions are satisfied: a + b > c a + c > b b + c > a

17.   Let b an unknown side of a triangle. To find the range of possible measure of side b, the inequality below may be used: (a- c) < b < (a + c)

18. Example 3: In Δ FIT, find the range of the possible lengths of   Let a = 5 and b = 8   a + b > c a + c > b b + c > a 5 + 8 > c 5 + c > 8 8 + c > 5       c < 13 c < 3 c < -3       (disregard because lengths must be positive) I F T 5 8 ? (Note: The figure is not drawn to scale)

19. Example 3: In Δ FIT, find the range of the possible lengths of     3 < c < 13 (The length of is greater than 3 and less than 13)   I F T 5 8 ? (Note: The figure is not drawn to scale)

20. Example 4: Which length/s can form a triangle? A.) 3, 4, 6 B.) 5, 6, 11 C.) 2, 3, 9 Solution: Check the lengths given if it will form a triangle using Triangle Inequality Theorem. a + b > c a + c > b b + c > a decision length 3 + 4 > 6 3 + 6 > 4 4 + 6 > 3 Triangle A. a=3, b=4, c=6 True True True

21. Example 4: Which length/s can form a triangle? A.) 3, 4, 6 B.) 5, 6, 11 C.) 2, 3, 9 Solution: Check the lengths given if it will form a triangle using Triangle Inequality Theorem. a + b > c a + c > b b + c > a decision length 5 + 6 > 11 5 + 11 > 6 6 + 11 > 5 Not a Triangle B. a=5, b=6, c=11 False True True

22. Example 4: Which length/s can form a triangle? A.) 3, 4, 6 B.) 5, 6, 11 C.) 2, 3, 9 Solution: Check the lengths given if it will form a triangle using Triangle Inequality Theorem. a + b > c a + c > b b + c > a decision length 2 + 3 > 9 2 + 9 > 3 3 + 9 > 2 Not a Triangle C. a=2, b=3, c=9 False True True

23. Alternative Solution: A.) 3, 4, 6

24. Alternative Solution: A.) 3, 4, 6

25. Alternative Solution: A.) 3, 4, 6

26. Alternative Solution: A.) 3, 4, 6

27. Alternative Solution: A.) 3, 4, 6

28. Alternative Solution: A.) 3, 4, 6

29. Alternative Solution: B.) 5, 6, 11

30. Alternative Solution: B.) 5, 6, 11

31. Alternative Solution: B.) 5, 6, 11

32. Alternative Solution: B.) 5, 6, 11

33. Alternative Solution: B.) 5, 6, 11

34. Alternative Solution: B.) 5, 6, 11

35. Alternative Solution: C.) 2, 5, 9

36. Alternative Solution: C.) 2, 5, 9

37. Alternative Solution: C.) 2, 5, 9

38. Alternative Solution: C.) 2, 5, 9

39. Alternative Solution: C.) 2, 5, 9

40. Alternative Solution: C.) 2, 5, 9

41. Alternative Solution: C.) 2, 5, 9

42. Alternative Solution: C.) 2, 5, 9

43. Alternative Solution: Based on the figures, only option A formed a triangle. Option B formed a straight line instead of a triangle and option C is short of sides that it cannot form a triangle. If we compare the sum of the two sides of options A, B, and C, only option A satisfied the conditions of Triangle Inequality Theorem.

44. a. 3 ft, 6 ft and 9 ft 3 + 6 > 9 b. 5 cm, 7 cm and 10 cm 5 + 7 > 10 7 + 10 > 5 5 + 10 > 7 c. 4 in, 4 in and 4 in Equilateral: 4 + 4 > 4 Is it possible for a triangle to have sides with the given lengths? Explain. (YES) (NO) (YES)

45. a. 6 ft and 9 ft 9 + 6 > x, x < 15 x + 6 > 9, x > 3 x + 9 > 6, x > – 3 15 > x > 3 5 cm and 10 cm 14 in and 10 cm Solve for the length of an unknown side ( X ) of a triangle given the lengths of the other two sides. The value of x: a + b > x > |a - b| 15 > x > 5 24 > x > 4

46.   The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.

47. Example 5: In the figure below, if m∠L = 45° and m∠O =105°, then by Exterior Angle. Inequality Theorem: 𝑚∠𝑂𝑉𝐸 > 𝑚∠𝐿 𝑚∠𝑂𝑉𝐸 > 𝑚∠𝑂 𝑚∠𝑂𝑉𝐸 > 45° 𝑚∠𝑂𝑉𝐸 > > 105°

48.   If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is longer than the third side of the second.

49.   Example 5: Compare the lengths of the third side of Δ BLU and Δ RED.

50. Given:       Solution:       Based on the angle-side theorem, the opposite side of the greater angle is longer side.

51.   If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second.

52. Are you familiar with hinge devices? Hinge devices are used to fasten or join two things together and allow adjustment, rotation, twisting or pivoting. The picture in the left is called hinge which is attached to our doors at home. It is responsible for the opening and closing of our doors. Scissors, compass and folding ladder are just some of the examples of hinge

53. Example 7: Which of the following pictures show a wide opening of the scissors?

54. Example 8: Given that ΔSAY and ΔTEL have two congruent sides as shown in the figure, which angle is greater, ∠A or ∠E?

55. Activity

56. I. Am I a Triangle? Directions: Which of the following could be the lengths of the sides of a triangle? Put a triangle ( ) if it forms a triangle and (X) if it does not form a triangle.   _______1. 1, 2, 3 _______2. 17, 16, 9 _______3. 9, 11, 18 ________4. 4, 8, 11 ________5. 5, 13, 6    

57. II. Directions: Modified True or False. Write True if the statement is correct, but if it’s False, change the underlined words to make it right. __________1. The measure of an exterior angle of a triangle is less than the measure of either remote interior angle. __________2. If one side of a triangle is longer than a second, then the angle opposite the first side is larger than the angle opposite the second side. greater true

58. __________3. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. __________ 4. If one angle of a triangle is larger than a second angle, then the side adjacent the first angle is longer than the side adjacent the second angle. __________5. If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is longer than the third side of the second. true opposite true