1. “Ratios and Proportions”

2. Ratio Ratio—compares two quantities in a fraction form with one number over another number. Proportion Proportion—two equal ratios.

3. To Solve Proportions “Cross-Multiply” ad = bc Multiply across from upper left to lower right and from upper right to lower left. What do you get? 8x = 5y

4. Ex #1: Solve for x in these Proportions “Cross-Multiply” 5x= 70 x = 14 “Cross-Multiply” 3x = 2(x + 1) 3x = 2x + 2 1x = 2 x = 2

5. Ex #2: Solve for x in this Proportion “Cross Multiply” 5(x – 7) = 2(x + 5) 5x – 35 = 2x + 10 3x – 35 = 10 3x = 45 x = 15

6. STUDY FOR UNIT 4 TEST (Friday, Dec. 14 th or Monday, Dec. 17th)  Triangles  Congruent Polygons  Congruent Triangles  SSS, SAS, ASA, AAS Problems  SSS, SAS, ASA, AAS Proofs  CPCTC and CPCTC Proofs  HL Theorem  Equilateral, Isosceles, and Right Triangle Problems and Solving for Missing Variables

7. Problem of the Day Problem of the Day Return Test and Collect SOL HW #5 “Angle 1 and Angle 2 are Supplementary Angles. m<1 = 12x + 8 and m<2 = 8x + 12. Find the m<2?” A. 20 degrees B. 28 degrees C. 104 degrees D. 76 degrees

8. Section 7.2 “Similar Polygons”

9. Ex I.: Ratio Problem What is the ratio of the length of the model to the length of the car? A Scale Model of a car is 4 in long. The actual car is 180 in long. What is the ratio of the length of the model to the length of the car? Model Car = 4 in Real Car180 in 1 in 45 in

10. Similar (~) and 1 Note Two figures that have the same shape but not necessarily the same size are similar. Similar Problems, Proportions 1. When solving Similar Problems, use Proportions.

11. To Tell if Similar Polygons?? 1. Corresponding Angles must be Equal. 2. Corresponding Sides must be in Proportional Ratios. Triangle ABC ~ RST B 50 50 10 20 10 20 85 45 22 85 45 A 22 C 11 11 45 85 T 45 85 R 1050 10 50 5 S

12. Ex #1: Complete each Statement ABCD ~ EFGH B C 53 A 53 D 127 F 127 G E H I. m<E = II. m<B = III. AB = AD EF ? IV. BC = AB ? EF V. GH = FG CD ?

13. Try Example #2 I. <H II. <R III. <X IV. HX V. HR

14. Ex #3: I. Polygons Similar (Yes/No)? II. Why w/Angles and Sides? I. B 1512 AC E18 16 20 D 24F II. Angles Equal B 1014 AC E15 18 12 D 20F

15. Try Example #4 I. No, Sides are Not Proportional Ratios II. Yes, Angles Equal and Sides are ¾ Ratio

16. Ex #5: For both, Find x and y w/Similar Figures LMNO ~ QRST x T x S 2 O 2 N Q R15 6 L 6 M ABC ~ YWZ Y 40 40 20 A20 y x y x W Z 30 B C3012

17. Try Example #6 I. x = 6 and y = 5 II. x = 9

18. Worksheet “Similar Polygons”

19. Problem of the Day #1 Problem of the Day #1 **Check Homework** 1. Yes, by 1/2 Ratio 2. No, Angles are not Equal 3. x = 4 4. x = 20 and y = 8

20. Problem of the Day #2 Problem of the Day #2 **Show Grades or Show Homework/Return Work** 1. x = 5.3 and y = 36 2. I. x = 7 II. BC = 11 III. MN = 22

21. “Proving Triangles Similar”

22. Try—3pts + 2pt EC Correct Find the Height of the Tree? 2m 3m 30m

23. Example #1 “A Small Child is 3 feet tall and is standing 6 feet from a flag pole. Another person is standing 12 feet from the same flag pole. How tall is that other person?” [Draw Picture]

24. 3 Similarities Angle-Angle (AA~) Similarity 1. Angle-Angle (AA~) Similarity “If Two Angles Congruent, then Triangles are Similar.” Side-Angle-Side (SAS~) Similarity 2. Side-Angle-Side (SAS~) Similarity “If Two Sides are Proportional and the middle Angle is Congruent, then Triangles are Similar.” Side-Side-Side (SSS~) Similarity 3. Side-Side-Side (SSS~) Similarity “If all three Sides are Proportional, then Triangles are Similar.”

25. One More Note AA~ Little Triangle Inside Big Triangle, then AA~.

26. Ex #2a: Explain why these Triangles are Similar? Write a Similarity Statement? W T 70 70 70 70 R M L <WMR = <TML (Vertical Angles), so AA~ Triangle WMR ~ Triangle TML by AA~

27. Ex #2b: Explain why these Triangles are Similar? Write a Similarity Statement? GR12 M 2418 F 16 H K <F = <M are Right Angles 12/16 = ¾ 18/24 = ¾ SAS~ Triangle FGH ~ Triangle MKR by SAS~

28. Ex #2c: Explain why these Triangles are Similar? Write a Similarity Statement? H T 1010 12 12 G C N M 2530 10/12 = 5/6 25/30 = 5/6 SSS~ Triangle GHC ~ Triangle NTM by SSS~

29. Ex #3: Explain why Similar? Then, find x? A 1012 10 12 B C D 18 X E SSS~ Postulate 12 = 10 18 X 12X = 180 X = 15 X 16 4 12 4 12 SAS~ Postulate X = 4 12 16 16X = 48 X = 3

30. Try Examples #4 I. AA~ or SAS~ and x = 9 II. SAS~ and x = 90

31. Ex #5: Explain why Similar? Then, find x? 6 9 2X AA~ Postulate 6 = 6 + 28 9 X 6X = 72 X = 12 4 5 X 15 15 AA~ Postulate 4 = X + 4 5 15 5X + 20 = 60; 5X = 40 X = 8

32. EXIT SLIP Worth: +10 Points EXIT SLIP Worth: +10 Points **COLLECT** Explain why Similar? Then, find x? I.A 84 8 4 B C D 6 X E II.90 110 11090X

33. 1. Worksheet “Similarity in Triangles” 2. SHORT QUIZ (Next Block)  Similar Polygons  Similar Polygon Problems  Similarity in Triangles (AA~, SAS~, and SSS~)  Similarity in Triangle Solving Problems

34. Problem of the Day Problem of the Day **Check Worksheet and Then Quiz** I. Explain why Similar (AA~, SAS~, SSS~)? II. Find x? 3 7 9 X

35. Then New Notes

36. “Proportions in Triangles” ‘2’ Theorems

37. 1. Triangle-Angle-Bisector Theorem “If an angle bisector bisects a triangle, then it divides the opposite side into two segments that are proportional to the triangle sides.” A B CDCD

38. Ex #1: Find x w/Triangle-Angle-Bisector A 6 6 X X B 5 D 5 8 C 8 5x = 48 x = 9.6 x = 9.6 8 8 5 x 3 5x = 24 x = 4.8

39. Ex #2: Find x w/Triangle-Angle-Bisector A 4 4 6 6 B D x C x 6 x = 3.6 x = 3.6 8 8 6 x 10 x = 5.7 x = 5.7

40. 2. Side-Splitter Theorem “If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides in proportions.” A c B C a d b D E

41. Ex #3: Find x w/Side-Splitter Theorem I. T x 5 x 5 SUSU 1610 R V Plug in values: 10x = 80 x = 8 II. x + 105 x + 105 x 3 x 3 3x + 30 = 5x 30 = 2x 15 = x

42. Try Ex #4: Find x and y? 67 x 14 9 y X = 12 and Y = 10.5

43. + 2pt EC Correct Solve “A man who is 6 feet tall casts a shadow that is 4 feet long. At the same time, a nearby flagpole casts a shadow that is 14 feet long. How tall is the flagpole?”

44. 1. Worksheets “2 Theorems” Test—Unit 5 2. Test—Unit 5 (Friday, January 11 th 1 st, 5 th, 7 th Blocks Monday, January 14 th 2 nd and 6 th Blocks)  Ratio and Proportion Problems  Scale Model w/Ratio Problems  Cross-Multiply to Find “x” Problems  Similar Polygons and their Problems  Similarity in Triangles (AA~, SAS~, and SSS~)  Similarity in Triangle Solving Problems  Triangle Angle Bisector and Side-Splitter Theorems  Triangle Inequality Problems

45. Problem of the Day Find x for Both? Problem of the Day Find x for Both? **Check Worksheet** 4 2 X 6 x = 12 x + 6 8 x – 2 2 X = 4.7 X = 4.7

46. Inequalities in Triangles ‘3’ Theorems

47. 1 st ‘2’ Theorems If an Unequal, (1) the longest side is across from the largest angle and (2) the largest angle is across from the longest side. If <A is largest, then BC is the longest side. B C A then)

48. List the Angles in Size from Smallest to Largest Ex #1: List the Angles in Size from Smallest to Largest K 21 ft38 ft 36 ft L 36 ft M Since Side KL is the smallest, <M then <K then <L (Greater then)

49. From a Triangle, List Angles from biggest to smallest Ex #2: From a Triangle, List Angles from biggest to smallest In Triangle ABC with AB = 12 ft AC = 11 ft BC = 8 ft [Hint: Draw the Triangle] Biggest to Smallest Angles <C to <B to <A

50. 1. Find x & 2. Find the Shortest and Longest Sides? Ex #3: 1. Find x & 2. Find the Shortest and Longest Sides? I. NII. H x x 64 64 C 8257 32 82 B 57 32 K V x = 34 degreesx = 91 degrees Shortest Side = CVShortest Side = BH Longest Side = NCLongest Side = BK (Greater 50 then) (Greater then)

51. From a Triangle, List Sides from shortest to longest Ex #4: From a Triangle, List Sides from shortest to longest In Triangle ABC with <A = 90 degrees <B = 40 degrees <C = 50 degrees [Hint: Draw the Triangle] Short to Long Sides AC, AB, to BC

52. 3 rd Theorem w/Wooden Popsicle Stick Activity (+8pts) 1. Cut the first Popsicle Stick into 1 in., 1 in., and 4 in. pieces. 2. Try to make a Triangle out of these three cut-out pieces. Glue them down on a piece of paper. 3. Cut the second Popsicle Stick into 2 in., 2 in., and 2 in. pieces. 4. Try to make a Triangle out of these three cut-out pieces. Glue them down on a piece of paper. Answer these Questions: 1. Which makes a Triangle (1 st or 2 nd Popsicle Stick)? 2. Why?? Inequality Theorem: YES NO If the sum of the lengths of a triangle is greater then the third side, then YES a triangle. If not, NO a triangle.

53. YES/NO Ex #5: For all Three A Triangle (YES/NO)? I. Sides 3 ft, 7ft, 8ft II. Sides 3cm, 6cm, 1ocm 3 + 6 > 10 (No) 3 + 7 > 8 (Yes) 7 + 8 > 3 (Yes) 3 + 8 > 7 (Yes) NO, not a Triangle YES, a Triangle III. Sides 1 ft, 9 ft, and 9ft YES, a Triangle YES, a Triangle

54. Example #6 Which of these three lengths COULD NOT be the lengths of the sides of a triangle? WHY?? A 7 m, 9 m, 5 m B 3 m, 6 m, 9 m C 5 m, 7 m, 8 m

55. Try Example #7 Which of these three lengths can COULD be the lengths of the sides of a triangle? WHY?? A 3 m, 14 m, 17 m B 11 m, 8 m, 12 m C 2 m, 3 m, 7 m

56. Example #8 Find the Possible Length of the Third Side, TK? T 35 ft K45 ft H A.10 < x < 45 B.35 < x < 80 C.10 < x < 80 D.10 < x < 80 (Greater then)

57. Try Example #9 Find the Possible Length of the Third Side, TK? T 25 ft K40 ft H A.25 < x < 65 B.15 < x < 40 C.15 < x < 65 D.15 < x < 65 (Greater then)

58. 1. Worksheets “Triangle Inequalities” 2. Test—Unit 5 (Friday, January 11 th 1 st, 5 th, 7 th Blocks Monday, January 14 th 2 nd and 6 th Blocks)  Ratio and Proportion Problems  Scale Model w/Ratio Problems  Cross-Multiply to Find “x” Problems  Similar Polygons and their Problems  Similarity in Triangles (AA~, SAS~, and SSS~)  Similarity in Triangle Solving Problems  Triangle Angle Bisector and Side-Splitter Theorems  Triangle Inequality Problems

59. Problem of the Day **Check HW and Then Test** SOL TEI Review Question Worth: +10pts 1. A 2. B 3. C 4. C 5. D 6. B 7. A 8. A 9. x = 80 degrees and BD (Longest Side) 120105 30130 7580

60. 1. SOL Homework #6 2. Semester Review Booklet 3. Extra Credit Sheet