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Properties of Congruent Triangles

Properties of Congruent Triangles. Congruence. Are the following pairs of figures the same?. They are the same!. Figures having the same shape and size are called congruent figures. Congruent Triangles and their Properties.

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Properties of Congruent Triangles

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  1. Properties of Congruent Triangles

  2. Congruence Are the following pairs of figures the same? They are the same! Figures having the same shape and size are called congruent figures.

  3. Congruent Triangles and their Properties If two triangles have the same shape and size, they are called congruent triangles. X A B Y C Z For the congruent triangles △ABCand △XYZ above, AB = XY, BC = YZ, CA = ZX A = X, B = Y, C = Z

  4. A X Y B C Z A and X, B and Y, C and Z are called corresponding vertices. AB and XY, BC and YZ, CA and ZX are called corresponding sides. A and X, B and Y, C and Z are called corresponding angles.

  5. A =X, B = Y, C = Z CA = ZX The corresponding vertices of congruent triangles should be written in the same order. In the above example,we can also write △BAC △YXZ, but NOT △CBA △XYZ. • All their corresponding sides are equal.  AB = XY, BC = YZ, X A B Y C Z The properties of congruent triangles are as follows: (i) All their corresponding angles are equal,  △XYZ △ABC is congruent to

  6. X Y Z If △ABC △XYZ … A 4.5 cm 4 cm 40° B C According to the properties of congruent triangles, △  △ B C X Y Z A AB = XZ AC = 4 cm XY = 4.5 cm =  Y B = = 40°

  7. A Q z cm x cm 7 cm R y P 130° 9 cm 30° C 4 cm = x 9 B Ð = Ð P A = - Ð - Ð o y 180 B C = - - o o o 180 130 30 = o 20 = QR BC z = 4 Follow-up question 1 In the figure, △ABC △PQR. Find the unknowns. According to the properties of congruent triangles, AC=PR

  8. Example 1 In the figure, AB = 5 cm, AC = 4 cm and BC = 7 cm. If△ABC@△DFE, find DE, EF and DF. Solution

  9. Example 2 In the figure, AB = 7 cm,∠A = 50° and ∠B = 30°. If △ABC@△PRQ, find PR, ∠P and ∠R. Solution

  10. Conditions for Congruent Triangles

  11. X A Y B Z C But, can we say that two triangles are congruent when only some of the properties of congruence are satisfied? Are these two triangles congruent? Yes… let’s see the following 5 conditions for congruent triangles first. Yes, because AB = XY, BC = YZ, CA = ZX, ∠A =∠X, ∠B =∠Y, ∠C =∠Z.

  12. C A B Z X Y Condition I SSS In △ABC and △XYZ, if AB = XY, BC = YZ and CA = ZX, then △ABC △XYZ. [Abbreviation: SSS]

  13. T F 2 cm 2 cm V D 4 cm 4 cm 5 cm 5 cm E U For example, TU = FE, UV = ED, TV = FD △TUV △FED (SSS)

  14. C A B Z X Y Condition II SAS In △ABC and △XYZ, if AB = XY, AC = XZ and A = X, then △ABC △XYZ. [Abbreviation: SAS] Note that ∠A and ∠X are the included angles of the 2 given sides.

  15. For example, T D 2.5 cm F U 2.5 cm 120° 120° 2 cm 2 cm V E UV= FE, TV = DE, V = E △TUV △DFE (SAS)

  16. A 3 cm E 3 cm C 2.5 cm 2.5 cm F 3.2 cm B 3.2 cm G Follow-up question 2 Determine whether each of the following pairs of triangles are congruent and give the reason. (a) △ABC △EGF (SSS) ◄AB = EG, BC = GF, AC = EF

  17. I N 2.5 cm L 50° 2 cm 2 cm K 50° 2.5 cm J M (b) △IJK △MNL (SAS) ◄IJ = MN, ∠J = ∠N, JK = NL

  18. Example 3 Are △MNP and △YZX in the figure congruent? If they are, give the reason. Solution Yes, △MNP△YZX. (SSS)

  19. Example 4 In the figure, WX = WY and ZX = ZY. Are △WXZ and △WYZ congruent? If they are, give the reason. Solution Yes, △WXZ△WYZ. (SSS)

  20. Example 5 Which two of the following triangles are congruent? Give the reason. Solution △PQR△WUV (SAS)

  21. Example 6 In the figure, AB = CD = 8 cm and ∠ABD = ∠CDB = 30°. Are △ABDand △CDB congruent? If they are, give the reason. Solution Yes, △ABD△CDB. (SAS)

  22. Example 7 Which two of the following triangles are congruent? Give the reason. Solution △DEF△ZYX (ASA)

  23. C Z A B X Y Condition III ASA In △ABC and △XYZ, if A = X, B = Y and AB = XY, then △ABC △XYZ. [Abbreviation: ASA] Note that AB and XY are the included sides of the 2 given angles.

  24. T 4 cm F D 20° 130° 130° 20° V U 4 cm E For example, U = F, UV = FD, V = D △TUV △EFD (ASA)

  25. In △ABC and △XYZ, if A = X, B = Y and AC = XZ, then △ABC △XYZ. [Abbreviation: AAS] C A B Z X Y Condition IV AAS Note that AC and XZ are the non-included sides of the 2 given angles.

  26. For example, T F D 20° 7 cm 130° 130° 7 cm 20° V U E U = F, V = D, TV = ED △TUV △EFD (AAS)

  27. A 5.25 cm G E 40° 45° 45° 40° B C 5.25 cm F Follow-up question 3 In each of the following, name a pair of congruent triangles and give the reason. (a) △ABC △FEG (ASA) ◄∠B = ∠E, BC = EG, ∠C = ∠G

  28. A 20° I L N 20° 100° 100° 12 cm J 12 cm 12 cm 20° C M 100° K B (b) △IJK △MNL (AAS) ◄∠J = ∠N, ∠K = ∠L, IK = ML

  29. Example 8 In the figure, ∠BAD = ∠CAD and AD⊥BC. Are △ABD and △ACD congruent? If they are, give the reason. Solution Yes, △ABD△ACD. (ASA)

  30. Example 9 Which two of the following triangles are congruent? Give the reason. Solution △PQR△ZYX (AAS)

  31. Example 10 In the figure, ∠ABC = ∠CDA and ∠ACB = ∠CAD. Are △ABCand △CDA congruent? If they are, give the reason. Solution Yes, △ABC△CDA. (AAS)

  32. A X B Y C Z Condition V RHS In △ABC and △XYZ, if C = Z = 90°, AB = XY and BC = YZ (or AC = XZ), then △ABC △XYZ. [Abbreviation: RHS]

  33. T F D 5 cm 2 cm 2 cm 5 cm V E U For example, U = F = 90°, TV = ED, TU = EF △TUV △EFD (RHS)

  34. D 6 cm A C B 6 cm Follow-up question 4 Are there any congruent triangles? Give the reason. Yes, △ABC △ADC. (RHS) ◄∠B = ∠D = 90°, AC = AC, BC = DC

  35. To sum up, two triangles are said to be congruent if any ONE of the following FIVEconditions is satisfied. C C C C X A A A A A B B B B 1. SSS 4. AAS Z Z Z Z 5. RHS X X X X Y Y Y Y 2. SAS B Y Z C 3. ASA

  36. Example 11 Are △ABC, △RPQ and △XYZ in the figure congruent? If they are, give the reasons. Solution △ABC△RPQ (RHS) △XYZ △RPQ (SAS) ∴△ABC, △RPQ and △XYZ are congruent.

  37. Example 12 In the figure, AB⊥BC, DC⊥BC and AC = DB. Are △ABC and △DCB congruent? If they are, give the reason. Solution Yes, △ABC△DCB. (RHS)

  38. Properties of Similar Triangles

  39. Similarity The following pairs of figures have the same shape, they are called similar figures. Similar figures have the same shape but not necessarily the same size.

  40. BC YZ CA ZX AB XY = = Similar Triangles and their Properties If two triangles have the same shape, they are called similar triangles. X A B C Z Y For the similar figures △ABC and △XYZ above, B = Y, C = Z A = X,

  41. X A Y B C Z A and X, B and Y, C and Z are called corresponding vertices. AB and XY, BC and YZ, CA and ZX are called corresponding sides. A and X, B and Y, C and Z are called corresponding angles.

  42. AB XY CA ZX BC YZ = = B = Y, C = Z A = X, Note: The corresponding vertices of congruent triangles should be written in the same order. • All their corresponding sides are proportional. X A B C Z Y The properties of similar triangles are as follows: (i) All their corresponding angles are equal, △ABC is similar to ~ △XYZ

  43. A 4.5 cm 4 cm 40° B C Ð B = o 40 X Y XY 2 cm 2 cm = 4 . 5 cm 4 cm Z = 2 . 25 cm If △ABC~ △XYZ ... According to the properties of similar triangles, Ð = Y XY XZ = ◄XZ and AC are correspondingsides. AB AC △ ~ △ B C X Y Z A

  44. Q z cm A 4 cm y R 10 cm P 5 cm AB AC QR PR x cm = = 132° 25° PQ PR BC AC C 4 cm z 5 B x 10 = = 4 10 4 5 = 2 = z x 8 Ð = Ð P A = - Ð - Ð o y 180 B C o o o = - - 180 132 25 o = 23 Follow-up question 5 In the figure, △ABC~ △PQR. Find the unknowns. According to the properties of similar triangles,

  45. Example 13 In △ABC and △RQP, BC = 1 cm, PQ = 2 cm, QR = 5 cm and PR = 4 cm. If △ABC ~ △RQP, find AB and AC.

  46. Solution

  47. Example 14 In the figure, AD = 3 cm, AC = 2 cm, CE = 4 cm, ∠A = 60° and BC⊥AE. If △ABC ~ △AED, find ∠E and AB.

  48. Solution

  49. Conditions for Similar Triangles

  50. A • All their corresponding angles are equal. • All their corresponding sides are proportional. B C X Y Z Two triangles are similar if any one of the following three conditions is satisfied. Conditions for Similar Triangles We have learnt that if two triangles are similar, then

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