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Chapter 10 Congruent and Similar Triangles

Chapter 10 Congruent and Similar Triangles. Introduction.

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Chapter 10 Congruent and Similar Triangles

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  1. Chapter 10Congruent and Similar Triangles

  2. 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.

  3. 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?

  4. 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.

  5. 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!

  6. 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 : 

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

  8. ? 65o 30o 5 cm 60o 90o ? ? THE ANGLE MEASURES OF A TRIANGLE AND CONGRUENT TRIANGLES • The sum of the angle measures of a triangle is 180o Example ? = 85o • Congruent triangles Congruent triangles are triangles with the same shape and size Angle = 60o; side = 5cm Example

  9. ? 82cm ? 52o 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 Example The side = 82 cm, the angle = 76o

  10. 60o ? 100cm ? Equilateral triangles • An equilateral triangle has three sides of equal length • In an equilateral triangle, the measure of each angle is 60o Example Angle = 60o, side = 100 cm

  11. Hypotenuse c Leg b 60o ? Leg a 3 cm ? 4 cm Right triangles and Pythagorean theorem • A right triangle is the triangle with one right angle • Pythagorean theorem c2 = a2 + b2 Example c2 = 42 + 32 = 25 C = 5

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

  13. 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 )

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

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

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

  17. B Example : E A C 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 ) D Textbook Page 44 Ex 10A Q 1 c, i

  18. A B C X Y Z 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) From given diagram, ACB = XZY (AAS)

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

  20. A Z C B X Y 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)

  21. B A C D 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

  22. Time to work • Class Work • Ex 10B Pg 49 • Q1 • Q2 • Q4 • Q6 • 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

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

  24. 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.

  25. Ratios and Similar Figures A B E F Example G H C D

  26. 14 m 7 m 3 m 6 m Ratios and Similar Figures Example These rectangles are similar, because the ratios of these corresponding sides are equal:

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

  28. 16 m n 10 m 5 m Proportions and Similar Figures You can use proportions of corresponding sides to figure out unknown lengths of sides of polygons. 10/16 = 5/n so n = 8 m

  29. ? 2cm 4 cm 65o 25o ? 12cm Similar triangles For two similar triangles, • Similar triangles are triangles with the same shape • corresponding angles have the same measure • length of corresponding sides have the same ratio Example Side = 6 cm Angle = 90o

  30. Similar Triangles 3 Ways to Prove Triangles Similar

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

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

  33. 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 25 degrees

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

  35. SSS Similarity Theorem If the sides of two triangles are in proportion, then the triangles are similar. D A C B F E

  36. 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. 14 21 18 12

  37. SAS Similarity Theorem D A C B F E 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.

  38. D A C B F E SAS Similarity Theorem Idea for proof

  39. Name Similar Triangles and Justify Your Answer!

  40. A 80 D E 80 B C ABC ~ ADE by AA ~ Postulate

  41. C 6 10 D E 5 3 A B CDE~ CAB by SAS ~ Theorem

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

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

  44. L 15 P A 25 9 N LNP~ ANL by SAS ~ Theorem

  45. Time to work !!!! • Class work Ex 10C Page 54 • Q2a to h • Q3 • Q5 • Q6 a to d • Q8 • Q10 • Q12 • Q13 • Home work Ex 10C Page 54 • Q1a to f • Q4 • Q7 • Q9 • Q11 • Q14 • Q15

  46. 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

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

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

  49. Home Work  Ex 10D Pg 62 • Q2 • Q6 • Q7 • Q11 • Q14 • Q17 • Q18

  50. 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

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