1. Deductive Geometry (I) 9 9.1Introduction to Deductive Reasoning 9.2Euclid and ‘Elements’ 9.3Deductive Proof about Lines and Triangles Chapter Summary Mathematics in Workplaces 9.4Deductive Proof about Congruent and Isosceles Triangles 9.5Deductive Proof about Similar Triangles

2. P. 2 Mathematics in Workplaces Lawyers use logic to analyze and present cases in courts. They reconstruct the sequence of events in criminal cases based on some known facts, which are called evidence. Lawyer The more evidence they gather, the more precise is the conclusion they can deduce.

3. P. 3 Do you recognize that we draw conclusions using different approaches? 9.1 Introduction to Deductive Reasoning 1.Intuitive approach Process of obtaining a conclusion by observation For example: One may conclude that lines A and B are non-parallel by direct observation. 2.Deductive reasoning Process of using logic to draw conclusions based on some known facts that are the premise For example: Premise: All apples in a box are rotten. Ray selects an apple from the box. Conclusion: The apple selected is rotten. Mathematicians show that deductive reasoning is an effective way to draw conclusions.

4. P. 4 A. Background of Euclid’s ‘Elements’ 9.2 Euclid and ‘Elements’ Euclid (about 330 – 275 B.C.) -a Greek mathematician -the ‘Father of Geometry’ -He wrote the most successful book of geometry in the history of mathematics called Elements.

5. P. 5 B. Introduction to Euclid’s ‘Elements’ 9.2 Euclid and ‘Elements’ In Elements, Euclid reorganized and summarized the mathematical knowledge from other mathematicians, and presented them in a systematic way. The Elements has 13 books which comprise a collection of definitions, axioms and theorems. A definition is the meaning of a term. For example, Definition 1: A point has no part. Definition 2: A line is breadthless length.

6. P. 6 B. Introduction to Euclid’s ‘Elements’ 9.2 Euclid and ‘Elements’ An axiom is a statement which assumes to be true without proof. For example, Axiom 3: We can describe a circle with a centre and a radius. Axiom 4: All right angles are equal. A theorem is logically true or proved. The proof is usually based on definitions, axioms or other theorems. For example, Theorem 17: In any triangle, the sum of any 2 angles is less than 2 right angles. Theorem 30: Straight lines parallel to the same straight line are also parallel to one another.

7. P. 7 B. Introduction to Euclid’s ‘Elements’ 9.2 Euclid and ‘Elements’ The figure shows the framework of the Elements. It shows a principle of construction: the later theorems built on top of the previous theorems, with definitions and axioms as the foundation stones. Theorem 3 Theorem 2 Theorem 1 Definition Axiom

8. P. 8 A. Angles Related to Intersecting Lines 9.3 Deductive Proof about Lines and Triangles Theorem 1: If AOB is a straight line, then a + b + c + d = 180°. (Reference: adj.  s on st. line) We will apply the above fact to prove the following theorem. Theorem 2: If AOB and COD are straight lines, then a = b. (Reference: vert. opp.  s) a + c = 180  adj.  s on st. line andb + c = 180  adj.  s on st. line  a + c = b + c a = b The reasons are presented in the right column. Proof:

9. P. 9 A. Angles Related to Intersecting Lines 9.3 Deductive Proof about Lines and Triangles Theorem 3: If a, b and c are angles at a point, then a + b + c = 360°. (Reference:  s at a pt.) a + c 1 = 180  adj.  s on st. line and b + c 2 = 180  adj.  s on st. line  a + b + c 1 + c 2 = 180  + 180  a + b + c = 360  Proof: Produce BO to D such that c is divided into c 1 and c 2.

10. P. 10 Example 9.1T In the figure, AOB, COD and EOF are straight lines.  AOC = 90 . Prove that a + b = 90 . Solution: A. Angles Related to Intersecting Lines 9.3 Deductive Proof about Lines and Triangles  AOE =  BOF = bvert. opp.  s  AOC +  AOE +  DOE = 180  adj.  s on st. line a + b = 90  90  + b + a = 180 

11. P. 11 Example 9.2T In the figure, AFB is a straight line. If  CFB =  AFD, prove that CFD is a straight line. Solution: A. Angles Related to Intersecting Lines 9.3 Deductive Proof about Lines and Triangles  AFD +  DFB = 180  adj.  s on st. line  CFB =  AFDgiven  CFB +  DFB = 180  CFD is a straight line.

12. P. 12 B. Angles Related to Parallel Lines 9.3 Deductive Proof about Lines and Triangles (a)Properties of Parallel Lines We will apply the previous fact to prove the following theorems. c = a vert. opp.  s a = b alt.  s, AB // CD  c = b Proof: Theorem 4: If AB // CD, then a = b. (Reference: alt.  s, AB // CD) Theorem 5: If AB // CD, then c = b. (Reference: corr.  s, AB // CD)

13. P. 13 B. Angles Related to Parallel Lines 9.3 Deductive Proof about Lines and Triangles (a)Properties of Parallel Lines a + d = 180  vert. opp.  s a = b alt.  s, AB // CD  b + d = 180  Proof: Theorem 6: If AB // CD, then b + d = 180 . (Reference: int.  s, AB // CD)

14. P. 14 Example 9.3T In the figure, AB // DE.  BAC = 150  and  EDC = 120 . Prove that AC  CD. Solution: B. Angles Related to Parallel Lines 9.3 Deductive Proof about Lines and Triangles Construct a line CF such that AB // CF // DE. a + 150  = 180  int.  s, AB // CF a = 30  b + 120  = 180  int.  s, CF // DE b = 60   ACD = a + b = 30  + 60  = 90  AC  CD

15. P. 15 B. Angles Related to Parallel Lines 9.3 Deductive Proof about Lines and Triangles (b)Tests for Parallel Lines and c = a vert. opp.  s that isa = b Proof: Theorem 7: If a = b, AB // CD. (Reference: alt.  s equal) Theorem 8: If c = b, AB // CD. (Reference: corr.  s equal) c = bgiven  AB // CD alt.  s equal

16. P. 16 B. Angles Related to Parallel Lines 9.3 Deductive Proof about Lines and Triangles (b)Tests for Parallel Lines c + d = 180  adj.  s on st. line b = c Proof:  AB // CD corr.  s equal Theorem 9: If b + d = 180 , AB // CD. (Reference: int.  s supp.) d = 180   c b + d = 180  given b + 180   c = 180 

17. P. 17 Example 9.4T In the figure, AGHB, CGD and EHF are straight lines.  AGC =  FHB = 120 . Prove that CD // EF. Solution: B. Angles Related to Parallel Lines 9.3 Deductive Proof about Lines and Triangles  DGH =  AGC = 120  vert. opp.  s  DGH =  FHB CD // EFcorr.  s equal

18. P. 18 Example 9.5T In the figure,  ABC = 25 , reflex angle ACD = 305  and  CDE = 30 . Prove that AB // DE. Solution: B. Angles Related to Parallel Lines 9.3 Deductive Proof about Lines and Triangles Construct a line FC such that AB // FC. a = 25  alt.  s, AB // FC  ACD = 360   305  s at a pt. = 55  a + b =  ACD b = 55   25  = 30   FCD =  CDE = 30  FC // DEalt.  s equal AB // DE

19. P. 19 C. Angles Related to Triangles 9.3 Deductive Proof about Lines and Triangles  a + b + c = 180  Proof: Theorem 10: In  ABC, a + b + c = 180 . (Reference:  sum of  ) Construct a line XAY such that XY // BC.  BAC +  XAB +  YAC = 180  adj.  s on st. line  XAB = b alt.  s, AB // FC  YAC = c alt.  s, AB // FC and

20. P. 20 C. Angles Related to Triangles 9.3 Deductive Proof about Lines and Triangles a + b = d Proof:  a + b + c = c + d a + b + c = 180   sum of  c + d = 180  adj.  s on st. line and Theorem 11: In  ABC, a + b = d. (Reference: ext.  s of  )

21. P. 21 Example 9.6T In the figure, EDB is a straight line.  ABD = 100 ,  EDC = 120  and  DCB = 40 . Prove that ABC is a straight line. Solution: C. Angles Related to Triangles 9.3 Deductive Proof about Lines and Triangles  EDC +  CDB = 180  adj.  s on st. line  CDB = 180   120  = 60  In  BCD,  CDB +  DCB +  CBD = 180  sum of   CBD = 80  60  + 40  +  CBD = 180   CBD +  ABD = 80  + 100  ABC is a straight line. = 180 

22. P. 22 A. Congruent Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles For a pair of congruent triangles, all pairs of corresponding sides and all pairs of corresponding angles are equal. If  ABC   XYZ, then 1.  A =  X,  B =  Y,  C =  Z;(corr.  s,  s) 2.AB = XY, BC = YZ, CA = ZX.(corr. sides,  s) We learnt the 5 conditions for congruent triangles: SSS, SAS, ASA, AAS and RHS. Now, let us discuss the deductive reasoning about congruent triangles.

23. P. 23 Example 9.7T In the figure, AD = AB and CD = CB. (a)Prove that  ABC   ADC. (b)Prove that  DCA =  BCA. Solution: A. Congruent Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles (a)In  ABC and  ADC, AC = ACcommon AB = ADgiven CB = CDgiven  ABC   ADCSSS (b)  ABC   ADCproved  DCA =  BCAcorr.  s,   s

24. P. 24 Example 9.8T In the figure,  BAE =  BCD and AB = BC. (a)Prove that  ABE   CBD. (b)Prove that DF = EF. Solution: A. Congruent Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles (a)In  ABE and  CBD,  BAE =  BCDgiven  ABE =  CBDcommon AB = BCgiven  ABE   CBDASA

25. P. 25 Example 9.8T In the figure,  BAE =  BCD and AB = BC. (a)Prove that  ABE   CBD. (b)Prove that DF = EF. Solution: A. Congruent Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles (b) AB = CB corr. sides,   s BD = BEcorr. sides,   s AD = CE AB  BD = CB  BE  BAE =  BCDgiven  AFD =  CFEvert. opp.  s  ADF   CEFAAS DF = EFcorr. sides,   s

26. P. 26 B. Isosceles Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles We will apply the properties of congruent triangles to prove the angles related to isosceles triangles. (a)Property of Isosceles Triangles Theorem 12: In  ABC, if AB = AC, then  B =  C. (Reference: base  s, isos.  )

27. P. 27 B. Isosceles Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles (a)Property of Isosceles Triangles Proof: Construct a line AD such that BD = CD. In  ABC and  ACD, AB = ACgiven AD = ADcommon side BD = CDby construction  ABD   ACDSSS  ABD   ACDcorr.  s,   s You may also deduce the properties of equilateral triangles from the theorem of isosceles triangles.

28. P. 28 Example 9.9T In the figure, AB = BC and  ACD =  DCB. Prove that  ADC = 3  ACD. Solution: B. Isosceles Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles AB = ACgiven  ABC =  ACBbase  s, isos   ACD =  DCBgiven  ABC =  ACB = 2  ACD =  ACD +  DCB In  BCD,  ADC =  ABC +  DCBext.  of  = 2  ACD +  ACD = 3  ACD

29. P. 29 B. Isosceles Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles (b)Test for Isosceles Triangles Proof: Construct a line AD such that  BAD =  CAD. In  ABD and  ACD,  ABD =  ACD given  BAD =  ACD by construction AD = ADcommon side  ABD   ACDAAS AB = ACcorr. sides,   s The following shows the converse of the theorem 12. We may also use the properties of congruent triangles to prove the theorem below. Theorem 13: In  ABC, if  B =  C, then AB = AC. (Reference: sides opp. eq.  s)

30. P. 30 Example 9.10T In the figure, AB = DB and  ADC = 90 . Prove that  BCD is an isosceles triangle. Solution: B. Isosceles Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles In  ABD, AB = DBgiven  BAD =  ADBbase  s, isos  In  ACD,  ACD +  CAD +  ADC = 180  sum of   ACD +  CAD + 90  = 180   ACD +  BAD = 90   ACD = 90    BAD

31. P. 31 Example 9.10T In the figure, AB = DB and  ADC = 90 . Prove that  BCD is an isosceles triangle. Solution: B. Isosceles Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles  BDC =  ADC   ADB = 90    ADB = 90    BAD  ACD =  BDC that is  BCD =  BDC BC = BDsides opp. eq.  s  BCD is an isosceles triangle.

32. P. 32 Example 9.11T In the figure, BD = CD and  ADB =  ADC. (a)Prove that  ADB   ADC. (b)Prove that  ABC =  ACB. Solution: B. Isosceles Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles (a) AD = ADcommon BD = CDgiven  ADB =  ADCgiven  ADB   ADCSAS

33. P. 33 Example 9.11T In the figure, BD = CD and  ADB =  ADC. (a)Prove that  ADB   ADC. (b)Prove that  ABC =  ACB. Solution: B. Isosceles Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles (b)  ABD =  ACDcorr.  s,   s BD = CDgiven  DBC =  DCBbase  s, isos   ABC =  ABD +  DBC  ACB =  ACD +  DCB  ABC =  ACB

34. P. 34 9.5 Deductive Proof about Similar Triangles For a pair of similar triangles, all pairs of corresponding angles are equal and all pairs of corresponding sides are proportional. If  ABC ~  XYZ, then 1.  A =  X,  B =  Y,  C =  Z; (corr.  s, ~  s) We learnt the 3 conditions for similar triangles: AAA, ratio of 2 sides, inc.  and 3 sides proportional. Now let us use these conditions to perform deductive reasoning about similar triangles. 2. (corr. sides, ~  s)

35. P. 35 9.5 Deductive Proof about Similar Triangles Example 9.12T In the figure, ABC and AED are straight lines.  AEB =  ADC. Prove that  ABE ~  ACD. Solution:  ABE =  ACD corr.  s, BE // CD  BAE =  CADcommon  ABE ~  ACD AAA  AEB =  ADCgiven BE // CDcorr.  s equal

36. P. 36 9.5 Deductive Proof about Similar Triangles Example 9.13T In the figure, ABD and CBE are straight lines. Prove that  ABE ~  DBC. Solution:  ABE =  DBCvert. opp.  s  ABE ~  DBCratio of 2 sides, inc. 

37. P. 37 9.5 Deductive Proof about Similar Triangles Example 9.14T In the figure, AB = 4, BC = 5, CD = 1, AD = DE = 4 and AE = 3.2.  ACB = 48 . (a)Prove that  ABC ~  AED. (b)Find  BAE. Solution: (a)In  ABC and  AED,  ABC ~  AED3 sides proportional

38. P. 38 9.5 Deductive Proof about Similar Triangles Example 9.14T In the figure, AB = 4, BC = 5, CD = 1, AD = DE = 4 and AE = 3.2.  ACB = 48 . (a)Prove that  ABC ~  ADE. (b)Find  BAE. Solution: (b)In  ABC, AC = 4 + 1 = 5 = BC  ABC =  BACbase  s, isos   ABC +  BAC +  ACB = 180  sum of   BAC +  BAC + 48  = 180  2  BAC = 132   BAC = 66   ABC ~  AEDproved  EAD =  BAC = 66  corr.  s, ~  s  BAE =  BAC +  EAD= 66  + 66  = 132 

39. P. 39 Chapter Summary 9.1 Introduction to Deductive Reasoning Deductive reasoning is the process of using logic to draw conclusion based on some known facts.

40. P. 40 Chapter Summary 9.2 Euclid and ‘Elements’ 2.The Elements has 13 books which comprise a collection of definitions, axioms and theorems. 1.Euclid reorganized and summarized the mathematical knowledge from other mathematicians and presented them in a systemic way in his book Elements.

41. P. 41 Chapter Summary 9.3 Deductive Proof about Lines and Triangles A. Angles Related to Intersecting Lines 1. If AOB is a straight line, then a + b + c + d = 180°. (Reference: adj.  s on st. line) 2. If AOB and COD are straight lines, then a = b. (Reference: vert. opp.  s) 3.If a, b and c are angles at a point, then a + b + c = 360°. (Reference:  s at a pt.)

42. P. 42 Chapter Summary 9.3 Deductive Proof about Lines and Triangles B. Angles Related to Parallel Lines 1. If AB // CD, then a = b. (Reference: alt.  s, AB // CD) 2. If AB // CD, then b = c. (Reference: corr.  s, AB // CD) 3. If AB // CD, then b + d = 180 . (Reference: int.  s, AB // CD) Properties of Parallel Lines

43. P. 43 Chapter Summary 9.3 Deductive Proof about Lines and Triangles B. Angles Related to Parallel Lines 1. If a = b, then AB // CD. (Reference: alt.  s equal) 2. If b = c, then AB // CD. (Reference: corr.  s equal) 3. If b + d = 180 , then AB // CD. (Reference: int.  s supp.) Tests for Parallel Lines

44. P. 44 Chapter Summary 9.3 Deductive Proof about Lines and Triangles C. Angles Related to Triangles 1. In  ABC, a + b + c = 180 . (Reference:  sum of  ) 2. In  ABC, a + b = d. (Reference: ext.  s of  )

45. P. 45 Chapter Summary 9.4 Deductive Proof about Congruent and Isosceles Triangles A. Congruent Triangles 5 conditions for congruent triangles: 1.SSS2.SAS3.ASA 4.AAS5.RHS

46. P. 46 Chapter Summary 9.4 Deductive Proof about Congruent and Isosceles Triangles B. Isosceles Triangles 1. If AB = AC, then  B =  C. (Reference: base  s, isos.  ) 2. If  B =  C, then AB = AC. (Reference: sides opp. eq.  s)

47. P. 47 Chapter Summary 9.5 Deductive Proof about Similar Triangles 3 conditions for similar triangles: 1.AAA 2.ratio of 2 sides, inc.  3.3 sides proportional

48. Follow-up 9.1 In the figure, AOB, COD and EOF are straight lines. Prove that a + b + c = 180 . Solution: A. Angles Related to Intersecting Lines 9.3 Deductive Proof about Lines and Triangles a + b + c = 180   BOD =  AOC = avert. opp.  s  FOD +  BOD +  BOE = 180  adj.  s on st. line c + a + b = 180 

49. Follow-up 9.2 In the figure, AOB is a straight line.  CFB = 45 . Prove that COD is a straight line. Solution: A. Angles Related to Intersecting Lines 9.3 Deductive Proof about Lines and Triangles adj.  s on st. line COD is a straight line.

50. Follow-up 9.3 In the figure, AB // DE,  BAC = 34  and  CDE = 56 . Prove that AC  CD. Solution: B. Angles Related to Parallel Lines 9.3 Deductive Proof about Lines and Triangles Construct a line FC such that BA // CF // ED. a = 34  alt.  s, BA // CF b = 56  alt.  s, CF // ED = 90   ACD = a + b = 34  + 56  AC  CD

51. Follow-up 9.4 In the figure, AOD and BOC are straight lines.  AOB = 70 ,  DOE = 45  and  DEO = 65 . Prove that BC // ED. Solution: B. Angles Related to Parallel Lines 9.3 Deductive Proof about Lines and Triangles  BOE +  AOB +  DOE = 180  adj.  s on st. line  BOE = 180   70   45  = 65   BOE =  OED = 65  BC // ED alt.  s equal

52. Follow-up 9.5 In the figure, OQT and OSV are straight lines.  PQT = 136 ,  QOS = 95  and  RSV = 129 . Prove that PQ // RS. Solution: B. Angles Related to Parallel Lines 9.3 Deductive Proof about Lines and Triangles Construct a line MO such that PQ // MO. a =  PQT = 136  corr.  s, PQ // MO b = 360   95   a  s at a pt. = 360   95   136  = 129   RSV = b = 129  RS // MOcorr.  s equal PQ // RS

53. Follow-up 9.6 In the figure, DBCE is a straight line.  DBA = 138  and  ACE = 112 . Prove that  ABC is an acute- angled triangle. Solution: C. Angles Related to Triangles 9.3 Deductive Proof about Lines and Triangles In  ABC,  ABC + 138  = 180  adj.  s on st. line  ABC = 42   ACB + 112  = 180  adj.  s on st. line  ACB = 68   ABC +  ACB +  BAC = 180  sum of   ABC is an acute-angled triangle.  BAC = 180   42   68  = 70 

54. Follow-up 9.7 In the figure, QRS is a straight line.  QPS =  RPS and QP = RP. (a)Prove that  QPS   RPS. (b)Prove that PS  QR. Solution: (a)In  QPS and  RPS, PQ = PRgiven  QPS =  RPSgiven PS = PS common A. Congruent Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles  QPS   RPSSAS

55. Follow-up 9.7 In the figure, QRS is a straight line.  QPS =  RPS and QP = RP. (a)Prove that  QPS   RPS. (b)Prove that PS  QR. Solution:  PSQ +  PSR = 180  adj.  s on st. line 2  PSR = 180   PSR = 90  A. Congruent Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles  PSQ =  PSRcorr.  s,   s PS  QR (b)  QPS   RPSproved

56. Follow-up 9.8 In the figure, AEC and BED are a straight lines. AB = AD and  AEB = 90 . (a)Prove that  ABE   ADE. (b)Prove that BC = CD. Solution: (a)In  ABE and  ADE,  AED = 180    AEBadj.  s on st. line = 180   90  A. Congruent Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles  ABE   ADERHS  AEB = 90  given AE = AE common = 90  AB = AD given

57. Follow-up 9.8 In the figure, AEC and BED are straight lines. AB = AD and  AEB = 90 . (a)Prove that  ABE   ADE. (b)Prove that BC = CD. Solution: (b)In  BEC and  DEC,  BEC = 180    AEBadj.  s on st. line A. Congruent Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles  BEC   DECSAS EC = ECcommon = 180   90  = 90   DEC = 180    AEDadj.  s on st. line = 180   90  = 90   BEC =  DEC BC = CDcorr. sides,   s BE = DEcorr. sides,   s

58. Follow-up 9.9 In the figure, PRS is a straight line and PQ = RQ.  QRS = 120 . Prove that  PQR is an equilateral triangle. Solution: B. Isosceles Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles In  PQR,  PRQ +  SRQ = 180  adj.  s on st. line  PRQ = 180   120  = 60   PRQ +  RPQ +  PQR = 180   sum of  60  + 60  +  PQR = 180   PRQ =  PQR =  RPQ = 60  PQ = RQgiven  RPQ =  PRQ = 60  base  s, isos   PQR = 60   PQR is an equilateral triangle.property of equil. 

59. Follow-up 9.10 In the figure,  CAB =  CBA and AB // PQ. Prove that  CPQ is an isosceles triangle. Solution: B. Isosceles Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles  CPQ =  CABcorr.  s, BA // QP  CQP =  CBAcorr.  s, BA // QP  CAB =  CBAgiven  CPQ is an isosceles triangle. CP = CQsides opp. eq.  s  CPQ =  CQP

60. Follow-up 9.11 In the figure, BE = CD, AB  CD and AC  BE. (a)Prove that  BCD   CBE. (b)Prove that  ABC is an isosceles triangle. Solution: B. Isosceles Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles (a)In  BCD and  CBE,  BCD   CBERHS BC = CBcommon BE = CDgiven AB  CD and AC  BEgiven  BCD and  CBE are right angled triangles.

61. Follow-up 9.11 In the figure, BE = CD, AB  CD and AC  BE. (a)Prove that  BCD   CBE. (b)Prove that  ABC is an isosceles triangle. Solution: B. Isosceles Triangles 9.4 Deductive Proof about Congruent and Isosceles Triangles  ABC is an isosceles triangle. AC = ABsides opp. eq.  s  DBC =  ECB corr.  s,   s (b)  BCD   CBEproved

62. Follow-up 9.12 In the figure, PRS and QRT are straight lines. PQ // TS. Prove that  PRQ ~  SRT. Solution: 9.5 Deductive Proof about Similar Triangles  RPQ =  RSTalt.  s, QP // ST  PRQ =  SRTvert. opp.  s  PRQ ~  SRTAAA In  PRQ and  SRT,  PQR =  STRalt.  s, QP // ST

63. Follow-up 9.13 In the figure, ADB and AEC are straight lines. Prove that  ABC ~  ADE. Solution: 9.5 Deductive Proof about Similar Triangles  BAC =  DAEcommon  ABC ~  ADEratio of 2 sides, inc. 

64. Follow-up 9.14 In the figure, ADC is a straight line. AD = BD = 5, CD = 4 and BC = 6,  CDB = 82 . (a)Prove that  ABC ~  BDC. (b)Find  BCD. Solution: 9.5 Deductive Proof about Similar Triangles (a)  ACB =  BCDcommon  ABC ~  BDCratio of 2 sides, inc. 

65. Follow-up 9.14 In the figure, ADC is a straight line. AD = BD = 5, CD = 4 and BC = 6,  CDB = 82 . (a)Prove that  ABC ~  BDC. (b)Find  BCD. Solution: 9.5 Deductive Proof about Similar Triangles (b) In  ABD, AD = BD = 5given  DBA = DAB base  s, isos   DBA +  DAB =  BDC ext.  of   DAB +  DAB = 82   DAB = 41   ABC ~  BDCproved  ABC =  CDB = 82  corr.  s, ~  s  ABC +  CAB +  BCD = 180  sum of 