2. Chapter 10 Congruent and Similar Triangles

3. Introduction Recognizing and using congruent and similar shapes can make calculations and design work easier. For instance, in the design at the corner, only two different shapes were actually drawn. The design was put together by copying and manipulating these shapes to produce versions of them of different sizes and in different positions. In this chapter, we will look in a little more depth at the mathematical meaning of the terms similar and congruent, which describe the relation between shapes like those in design.

4. Similar and Congruent Figures Congruent polygons have all sides congruent and all angles congruent. Similar polygons have the same shape; they may or may not have the same size. Worksheet : Exercise 1 : Which of the following pairs are congruent and which are similar?

5. Examples These figures are similar and congruent. They’re the same shape and size. These figures are similar but not congruent. They’re the same shape, but not the same size.

6. Another Example These figures are neither similar nor congruent. They’re not the same shape or the same size. Even though they’re both triangles, they’re not similar because they’re not the same shape triangle. Note: Two figures can be similar but not congruent, but they can’t be congruent but not similar. Think about why!

7. Congruent Figures When 2 figures are congruent, i.e. 2 figures have the same shape and size,  Corresponding angles are equal  Corresponding sides are equal  Symbol : 

8. Congruent Triangles AB = XY, BC = YZ, CA = ZX  A =  X,  B =  Y,  C =  Z A C B X Z Y Note : Corresponding vertices are named in order.

9. THE ANGLE MEASURES OF A TRIANGLE AND CONGRUENT TRIANGLES The sum of the angle measures of a triangle is 180 o Example 30 o 65 o ? ? = 85 o Congruent triangles 90 o 60 o 5 cm ? ? Example Congruent triangles are triangles with the same shape and size Angle = 60 o ; side = 5cm

10. Isosceles triangles An isosceles triangle is the triangle which has at least two sides with the same length In an isosceles triangle, angles that are opposite the equal-length sides have the same measure The side = 82 cm, the angle = 76 o 82cm 52 o ? ? Example

11. Equilateral triangles An equilateral triangle has three sides of equal length In an equilateral triangle, the measure of each angle is 60 o Example 60 o 100cm ? ? Angle = 60 o, side = 100 cm

12. Right triangles and Pythagorean theorem A right triangle is the triangle with one right angle Pythagorean theorem c 2 = a 2 + b 2 Example Leg a Leg b Hypotenuse c c 2 = 4 2 + 3 2 = 25 60 o ? 3 cm 4 cm ? C = 5

13. Ex 10A Page 47 Q2 a By comparing, x = 4.8, y = 42 Q2 d By comparing, x = 22, y = 39 – 22 = 17 Q2 b By comparing, x = 16, y = 30 ( 180  - 75  - 75  )

14. Tests for Congruency Ways to prove triangles congruent : SSS ( Side – Side – Side ) SAS ( Side – Angle – Side ) ASA ( Angle – Side – Angle ) or AAS ( Angle –Angle – Side ) RHS ( Right angle – Hypotenuse – Side )

15. SSS ( Side – Side –Side ) Three sides on one triangle are equal to three sides on the other triangle. A BC X Y Z AB = XY, BC = YZ, CA = ZX (SSS)

16. Example : Given AB = DB and AC = DC. Prove that  ABC   DBC AB = DB ( Given ) AC = DC ( Given ) BC ( common) Hence  ABC   DBC ( SSS ) A C B D Textbook Page 44 Ex 10A Q 1 a, k

17. SAS ( Side – Angle – Side ) Two pairs of sides and the included angles are equal. AB = XY, BC = YZ,  ABC =  XYZ ( included angle ) (SAS) A BC X Y Z

18. Example : Given AC = EC and BC = DC. Prove that  ABC   EDC AC = EC ( Given )  ACB =  ECD ( included angle, vert opp ) BC = DC ( Given ) Hence  ABC   EDC ( SAS ) Textbook Page 44 Ex 10A Q 1 c, i A B C D E

19. ASA ( Angle – Side – Angle ) AAS ( Angle – Angle – Side ) Two pairs of angles are equal and a pair of corresponding sides are equal. AB = XY,  ABC =  XYZ  BAC =  YXZ (ASA) A BC X Y Z From given diagram,  ACB =  XZY (AAS)

20. Example : Given AC = EC and  BAC =  DEC Prove that  ABC   DEC AC = EC ( Given )  BAC =  DEC ( Given )  ACB =  ECD (vert opp) Hence  ABC   EDC ( ASA ) Textbook Page 44 Ex 10A Q 1 f, o A B C D E

21. RHS ( Right angle – Hypotenuse – Side ) Right-angled triangle with the hypotenuse equal and one other pair of sides equal.  ABC =  XYZ = 90° ( right angle) AC = XZ ( Hypotenuse) BC = YZ (RHS) A B C X Y Z

22. Example : Prove that  ABC   DBC  ACB =  DCB = 90  AB = DB ( Given, hypotenuse ) BC is common Hence  ABC   EBC ( RHS ) Textbook Page 44 Ex 10A Q 1 g, j Try Q1 e, 1y too DA B C

23. Time to work Home Work Ex 10A Page 44- 47 Q 1 b, h, m, p, r, x Q 2 c, e Ex 10B Pg 49-50 Q3, 5, 7, 8 Class Work Ex 10B Pg 49 Q1 Q2 Q4 Q6

24. Thinking Time ????? If 3 angles on  A are equal to the 3 corresponding angles on the other  B, are the two triangles congruent ?

25. Ratios and Similar Figures Similar figures have corresponding sides and corresponding angles that are located at the same place on the figures. Corresponding sides have to have the same ratios between the two figures.

26. Ratios and Similar Figures Example A E C F D GH B These sides correspond: AB and EF BD and FH CD and GH AC and EG These angles correspond: A and E B and F D and H C and G

27. Ratios and Similar Figures Example 7 m 3 m 6 m 14 m These rectangles are similar, because the ratios of these corresponding sides are equal:

28. A proportion is an equation that states that two ratios are equivalent. Examples: n = 5 m = 4 Proportions and Similar Figures

29. You can use proportions of corresponding sides to figure out unknown lengths of sides of polygons. 16 m 10 m n 5 m 10/16 = 5/n so n = 8 m

30. Similar triangles Similar triangles are triangles with the same shape For two similar triangles, corresponding angles have the same measure length of corresponding sides have the same ratio 65 o 25 o ? 4 cm 2cm 12cm ? Example Angle = 90 o Side = 6 cm

31. Similar Triangles 3 Ways to Prove Triangles Similar

32. Similar triangles are like similar polygons. Their corresponding angles are CONGRUENT and their corresponding sides are PROPORTIONAL. 6 10 8 3 4 5

33. But you don’t need ALL that information to be able to tell that two triangles are similar….

34. AA Similarity If two angles of a triangle are congruent to the two corresponding angles of another triangle, then the triangles are similar. 25 degrees

35. SSS Similarity If all three sides of a triangle are proportional to the corresponding sides of another triangle, then the two triangles are similar. 18 12 8 14 21

36. SSS Similarity Theorem If the sides of two triangles are in proportion, then the triangles are similar. A BC D EF

37. SAS Similarity If two sides of a triangle are proportional to two corresponding sides of another triangle AND the angles between those sides are congruent, then the triangles are similar. 18 21 12 14

38. A If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar. BC D EF SAS Similarity Theorem

39. D EF A BC Idea for proof

40. Name Similar Triangles and Justify Your Answer!

41. A BC DE 80   ABC ~  ADE by AA ~ Postulate

42. AB C DE  CDE~  CAB by SAS ~ Theorem 6 3 10 5

43. O N L K M  KLM~  KON by SSS ~ Theorem 6 3 10 5 6 6

44. C B A D  ACB~  DCA by SSS ~ Theorem 24 36 20 30 16

45. N L A P  LNP~  ANL by SAS ~ Theorem 259 15

46. Time to work !!!! oClass work Ex 10C Page 54  Q2a to h  Q3  Q5  Q6 a to d  Q8  Q10  Q12  Q13 oHome work Ex 10C Page 54  Q1a to f  Q4  Q7  Q9  Q11  Q14  Q15

47. Areas of Similar Figures Activity : Complete the table for each of the given pairs of similar figures Conclusion: If the ratio of the corresponding lengths of two similar figures is then the ratio of their areas is

48. Thinking Time Does the identity works for the following figures ? Why?

49. Time to work !!! Class work Ex 10D Pg 62 Q 10 Q12 Q13 Q15 Q16 Q20 - 22 Class work Ex 10D Pg 62 Q1 a to d Q3 Q4 Q5 Q8 Q9

50. Home Work Ex 10D Pg 62 Q2 Q6 Q7 Q11 Q14 Q17 Q18

51. Volumes of Similar Solids Activity : Complete the table for each of the given pairs of similar Solids Conclusion: If the ratio of the corresponding lengths of two similar figures is then the ratio of their volumes is

52. Total Surface Area of similar solids If the ratio of the corresponding lengths of two similar figures is then the ratio of their total surface areas is

53. Time to work Class work Ex 10E Pg 67 Q1 Q2 Q3 Q4 Q5 Class work Ex 10E Pg 67 Q6 Q9 Q11

54. Q3 ? How to find the weight of a similar solid??? If both solids were made from the same material, Density will be the same Hence using the formula : Density = Mass  Volume

55. Your favourite moment Home work Ex 10E Pg 67 Q7 Q8 Q10 Q12 Q13 Q14