2. By : Darto, SMP N 4 Pakem Media and Video

5. Problem -2 Problem -1

6. Angles in a Triangles

7. PROBLEM -1 1.Determine the angle in the triangle isn’t known yet if given two angle : a. 23, 67, … b. 37, 84, … 2. Determine the value of x if the angle in the triangle are a. 4x, 5x+6, 9x -6 b. 2x, 3x, 5x

8. EXERCISES-1

9. EXERCISES-2

13. Conclusion

14. Pasangan seg. Garis(cm) Cek, please BisaTidak 4, 7, 8 5, 6, 10 7, 15, 6 8, 10, 19

15. 14 The sum of the lengths of any two sides of a triangle is greater than the third side. 512 15 5+15 >12 or 20>12

16. 15 The sum of the lengths of any two sides of a triangle is greater than the third side. 512 15 12+15 >5 or 27>5

17. 16 The sum of the lengths of any two sides of a triangle is greater than the third side. 512 15 5+12 >15 or 17>15 5+15 >12 or 20>12 12+15 >5 or 27>5

18. 17 The measures of two sides of a triangle are 15 and 8. Between what two numbers is the third side. X 15+8 > X 15+X > 8 8+X > 15 STANDARD 6 23 > X X < 23 15+X > 8 -15 X > -7 8+X > 15 -8 X > 7 0 5 10 15 20 25 x -5 -10 -15 - 20 xxx -7 7 23 X | 7<X<23 8 15 The third side will be any value between 7 and 23. 7 23

19. 18 If a triangle has sides of measure x, x+4, 3x-5, find all possible values of x (X+4)+(3X-5) > X (X+4 )+X > (3X-5) X X+4 3X-5 4X -1 >X -4X -1 >-3X -3.3 <X X>.3 2X +4 > 3X-5 -2X 4 > X-5 +5 9 > X X < 9 Sign (>) changes when dividing by (-3) 0 5 10 15 20 25 x -5 -10 -15 - 20 xxx 9 3.3 (3X-5) +X > (X+4 ) 4X – 5 > X +4 -X -X 3X – 5 > 4 +5 3X > 9 3 X > 3 3 9 X | 3<X< 9

20. 19 If one side of a triangle is the longest then A B C

21. 20 If one side of a triangle is the longest then The opposite angle to this side is the largest A B C

22. 21 And the angle opposite to the shortest side A B C

23. 22 The sum of the lengths of any two sides of a triangle is greater than the third side. 512 15 5+12 >15 or 17>15

24. 1. Do you remember about acute angle 2.Observe the size all of angle in the triangle bellow

25. 1. Do you remember about obtuse angle 2. Observe the size all of angle in the triangle bellow

26. THE KINDS OF TRIANGLE BASE ON THE SIZE ANGLE III.1. Do you remember about right angle 2. Observe the size all of angle in the triangle bellow

27. Problem -3 Determine the kind of triangle bellow if 1. The angle are : 65, 75, 80 2. The angle are : 25, 60, 95 3. The angle are : 54, 56, 70 4. Two angle are : 73, 34, 5. The proportion of angle is 3 : 4 : 5 6. The proportion of angle is 2 : 3 : 4 7. The angle is 6x, 2x + 3, 4x +9

28. PROBLEM-3

29. THE KINDS OF TRIANGLE BASE ON THE LENGTH SIDE I.1. Observe the length of all side in the triangle bellow

30. THE KINDS OF TRIANGLE BASE ON THE LENGTH SIDE II.1. Observe the length of all side in the triangle bellow

31. THE KINDS OF TRIANGLE BASE ON THE LENGTH SIDE III.1. Observe the length of all side in the triangle bellow

33. RIGHT TRIANGLES 1. Recall Pythagorean theorem 2. Indentify The kinds of Triangle by using Pythagorean theorem 3. The kinds of Triple Pythagorean number and its expectation 4. The specific side proportion of right triangle

34. 30°-60°-90° TRIANGLE 45°-45°-90° TRIANGLE PROBLEM 1 PROBLEM 2 PROBLEM 4 PROBLEM 5 PROBLEM 6 PROBLEM 3

35. 2 2 2 60° 30° 2 2 1 1 1. An equilateral triangle is also equiangular, all angles are the same. 2. Let’s draw an Altitude from one of the vertices. Which is also a Median and Angle bisector. 3. The bisected side is divided into two equal segments and the bisected angle has now two 30° equal angles. How is the right angle that was formed? Click to find out

36. 60° 30° 1 2 2 1 60° 30° 1 2 z 2 = z + 1 2 2 2 4 = z + 1 2 3 = z 2 2 z = 3 4. The triangle is divided into 2 right angles with acute angles of 30° and 60° 5. Let’s draw the top triangle and label the unknown side as z. 6. Let’s apply the Pythagorean Theorem to find the unknown side. Can we generalize this result for all 30°-60°-90° right triangles? Click to find out…

37. 60° 30° 1 2 3 (.5) (2) (s) 1 2 3 60° 30° 1 2 3 60° 30° 1 2 3 60° 30° 7. Is this true for a triangle that is twice as big? 8. Is this true for a triangle that is half the original size? 9. What about a triangle that is “s” times bigger or Smaller? Click to find out…

38. 2 3s s s 60° 30° 3 In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg. THEOREM 8-7

39. Find the values of the variables. Round your answers to the nearest hundredth. x y 30° 50 90°-30°=60° 60° 2 3s s s 30° 2x = 50 2 2 x =25 y = x3 y = 25 3 OR Is this 30°-60°-90°? Then we know that: y = 43.30

40. 60° 30° 90°-60°=30° x y 90 90 = x3 3 33 90 3 = x 3 3. 3 90 3 ( ) 2 3 90 3 = x 3 30 = x OR 2x = y 3 30 ( ) = y 2 3 60 y=3 60 OR 3 30 x= Find the values of the variables. Round your answers to the nearest unit. Is this a 30°-60°-90°? y =104 x = 52 2 3s s s 60° 30°

41. 60° 30° 90°-60°=30° x y 30 30 = x3 3 33 30 3 = x 3 3. 3 30 3 ( ) 2 3 30 3 = x 3 10 = x 2x = y 3 10 ( ) = y 2 3 20 y=3 20 3 10 x= Find the values of the variables. Find the exact answer. Is this a 30°-60°-90°? 2 3s s s 60° 30°

42. 1 1 1 1 1. Let’s draw a diagonal for the square above. The diagonal bisects the right angles of the square. What kind of right triangles are form? Click to find out…

43. 1 1 1 1 45° 1 1 y y = 1 + 1 222 2 y = 2 2 2 2. The triangles are 45°-45°-90° 3. Let’s draw the bottom triangle and label the hypotenuse as y 4. Let’s apply the Pythagorean Theorem to find the hypotenuse. Can we generalize our findings? Click to find out…

44. 45° 1 1 2 (.5) (1.5) s s s 1 1 2 45° 1 1 2 5. Let’s draw a triangle half the size of the original. 6. Let’s draw a triangle one and a half the size of the original. 7. Let’s draw a triangle S times the size of the original. Click to see our findings…

45. s s s2 In a 45°-45°-90° triangle, the hypotenuse is times as long as a leg. THEOREM 8-6 2 45°

46. 90°-45°=45° 45° x y 36 If y = x 36 = x2 2 22 36 2 = x 2 2. 2 36 2 ( ) 2 2 36 2 = x 2 18 = x 2 18 x = OR 2 18 y = OR then Find the values of the variables. Round your answers to the nearest tenth. s s s2 45° Is this a 45°-45°-90°? x = 25.5 y = 25.5

47. 45° 90°-45°=45° 45° x y 42 If y = x 42 = x2 2 22 42 2 = x 2 2. 2 42 2 ( ) 2 2 42 2 = x 2 21 = x 2 21 x = 2 21 y =then Find the values of the variables. Give an exact answer. s s s2 45° Is this a 45°-45°-90°?

48. 45° 90°-45°=45° x 21 y 45° x = 21 2 x y= 2 21 y = Find the values of the variables. Give the exact answer. Is this a 45°-45°-90°? s s s2 45°