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Congruence and Similarity

Congruence and Similarity. Form 1 Mathematics Chapter 11. Reminder. Lesson requirement Textbook 1B Workbook 1B Notebook Before lessons start Desks in good order! No rubbish around! No toilets! Keep your folder at home Prepare for Final Exam. Reminder. Missing HW Detention SHW (I)

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Congruence and Similarity

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  1. Congruenceand Similarity Form 1 Mathematics Chapter 11

  2. Reminder • Lesson requirement • Textbook 1B • Workbook 1B • Notebook • Before lessons start • Desks in good order! • No rubbish around! • No toilets! • Keep your folder at home • Prepare for Final Exam

  3. Reminder • Missing HW • Detention • SHW (I) • 8th May (Wednesday) • SHW (II) • 14th May (Tuesday) • OBQ • 15th May (Wednesday) • CBQ • 20th May (Monday)

  4. X Y Congruence (全等,p.172) • Congruent figures (全等圖形) 1. Figures having the same shape and size are called congruent figures. e.g. Figures X and Y below are congruent figures. 2.Two congruent figures can fit exactly on each other.

  5. Transformation and Congruence (p.174) • We learnt transformations in Chapter 9: • Translation • Reflection • Rotation • Enlargement and Reduction • Which one will NOT change the shape? • Try Class Discussion on p.174

  6. Transformation and Congruence (p.174) 1.When a figure is translated, rotated or reflected, the image produced is congruent to the original figure. 2. When a figure is reduced or enlarged, the image produced will not be congruent to the original one.

  7. A X C Z B Y Congruent Triangles(全等三角形,p.175) • Symbol “  ” means “is congruent to” When two triangles are congruent, (i)their corresponding sides (對應邊) are equal, (ii)their corresponding angles(對應角) are equal. e.g. If △ABC △XYZ, then AB = XY, BC = YZ, CA = ZX, A = X, B = Y, C = Z.

  8. Congruent Triangles (p.175) • Example 1: Name a pair of congruent triangles in the figure. From the figure above, we have △ABC  △RQP.

  9. Congruent Triangles (p.175) • Example 2: Given that △ABC  △XYZ, find the unknowns x, b and y. ∵ The corresponding sides and corresponding angles of two congruent triangles are equal, ∴ x = 5 cm, b = 6 cm, y = 50°

  10. Congruent Triangles (p.175) • Example 3: Since △PQS △RSQ, we have PQ = RS i.e. s = 8 cm and RQS = PSQ = 70° In △RSQ, 45° + RQS + r = 180° 45° + 70° + r = 180° 115° + r = 180° r = 65° Given that △PQS  △RSQ, find the unknowns s and r.

  11. Time for Practice • Page 176 of Textbook 1B • Class Practice • Pages 177 – 178 of Textbook 1B • Questions 4 – 17 • Pages 74 – 75 of Workbook 1B • Questions 2 – 5

  12. Conditions for Triangles to be Congruent (p.179) • There are four common conditions: • SSS: 3 Sides Equal • SAS: 2 Sides and Their Included Angle Equal • ASA : 2 Angles and 1 Side Equal(AAS) • RHS: 1 Right-angle, 1 Hypotenuses (斜邊) and 1 Side Equal

  13. 3 Sides Equal (SSS, p.179) If AB = XY, BC = YZ and CA = ZX, then △ABC  △XYZ. [Reference: SSS]

  14. 3 Sides Equal (SSS, p.179) • Example 1: Determine which pair of triangles in the following are congruent. The lengths of the three sides of (I) and (III) are 4, 6 and 8. ∴ (I) and (III) are congruent triangles. (SSS)

  15. 3 Sides Equal (SSS, p.179) • Example 2: Write down a pair of congruent triangles and give reasons. AB = CD (Given) BD = DB (Common side) DA = BC (Given) △ABD  △CDB (SSS)

  16. 2 Sides and Their Included Angle Equal (SAS, p.180) If AB = XY, B = Y and BC = YZ, then △ABC  △XYZ. [Reference: SAS]

  17. 2 Sides and Their Included Angle Equal (SAS, p.180) • Example 1: Determine which pair of triangles in the following are congruent. The lengths of the two sides of (I) and (III) are 5 and 6 and their included angles are both 40°. ∴ (I) and (III) are congruent triangles. (SAS)

  18. 2 Sides and Their Included Angle Equal (SAS, p.180) • Example 2: Write down a pair of congruent triangles and give reasons. CA = CE (Given) CB = CD (Given) ACB = ECD (Given) △ACB  △ECD (SAS)

  19. 2 Sides and Their Included Angle Equal (SAS, p.180) • Note: Must be SAS, not SSA! The abbreviation for this condition for congruent triangles is SAS, where the ‘A’ is written between the two ‘S’s to indicate an included angle. If we write SSA, then it means ‘two sides and a non-included angle’, but this is not a condition for congruent triangles. For example:

  20. 2 Angles and 1 Side Equal (ASA or AAS, p.181) If A = X,AB = XY and B = Y, then △ABC  △XYZ. [Reference: ASA] or If A = X,B = Y and BC = YZ, then △ABC  △XYZ. [Reference: AAS]

  21. 2 Angles and 1 Side Equal (ASA or AAS, p.181) • Example 1: Determine which pair(s) of triangles in the following are congruent. The two angles of (I) and (IV) are 45° and 70° while their included sides are both 8. ∴ (I) and (IV) are congruent triangles. (ASA) The two angles of (II) and (III) are 45° and 70° while the lengths of the sides opposite to 70° are both 8. ∴(II) and (III) are congruent triangles. (AAS)

  22. 2 Angles and 1 Side Equal (ASA or AAS, p.181) • Example 2: Write down a pair of congruent triangles and give reasons. BAD = CAD (Given) AD = AD (Common side) ADB = ADC (Given) △ADB  △ADC (ASA)

  23. 2 Angles and 1 Side Equal (ASA or AAS, p.181) • Example 3: Write down a pair of congruent triangles and give reasons. ABD = ACD (Given) BDA = CDA (Given) DA = DA (Common side) △BDA  △CDA (AAS)

  24. 1 Right-angle, 1 Hypotenusesand 1 Side Equal (RHS, p.183) If C = Z = 90°, AB = XY and BC = YZ, then △ABC  △XYZ. [Reference: RHS]

  25. 1 Right-angle, 1 Hypotenusesand 1 Side Equal (RHS, p.183) • Example 1: Determine which pair of triangles in the following are congruent. (I) and (III) are both right-angled triangles. Also, the hypotenuses and the sides of (I) and (III) are both 6 and 4 respectively. ∴ (I) and (III) are congruent triangles.

  26. 1 Right-angle, 1 Hypotenusesand 1 Side Equal (RHS, p.183) • Example 2: In the figure, CAD and ACB are both right angles and DC = BA. Determine whether △DAC and △BCA arecongruent and give reasons. DAC=  BCA= 90 DC = BA (Given) AC = CA (Common side) Yes, △DAC  △BCA (RHS).

  27. Conditions for Triangles to be Congruent (p.185) The table below summarizes all the conditions needed for two triangles to be congruent: SSS ASA AAS RHS SAS

  28. Time for Practice • Page 185 of Textbook 1B • Class Practice • Pages 186 – 187 of Textbook 1B • Questions 1 – 17 • Pages 76 – 79 of Workbook 1B • Questions 1 – 5

  29. Enjoy the world of Mathematics! Ronald HUI

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