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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 (II)

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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 (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) 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.

  6. 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.

  7. 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

  8. 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

  9. 3 Sides Equal (SSS, p.179) If AB = XY, BC = YZ and CA = ZX, then △ABC  △XYZ. [Reference: SSS]

  10. 2 Sides and Their Included Angle Equal (SAS, p.180) If AB = XY, B = Y and BC = YZ, then △ABC  △XYZ. [Reference: SAS]

  11. 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:

  12. 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]

  13. 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]

  14. 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

  15. 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

  16. Similarity (相似,p.188) • Similar figures (相似圖形) 1.Figures having the same shape are called similar figures. e.g. Figures A and B are similar figures. 2.When a figure is enlarged or reduced, the new figure is similar to the original one. Note: Two congruent figures always have the same shape, and so they must be similar figures.

  17. X A XY AB YZ BC ZX CA C B Z Y Similar Triangles (相似三角形,p.189) • Symbol “ ~ ” means “is similar to” When two triangles are similar, (i)their corresponding angles are equal, (ii)their corresponding sides are proportional. e.g. If △ABC~ △XYZ, then A = X, B = Y, C = Z, = = .

  18. 40z 2835 = 40  3528 As As = z ABPQ BCQR BCQR ACPR = = y40 2835 = 28  4035 y= Similar Triangles (p.189) • Example 1: In the figure, △ABC ~ △PQR. Find the unknowns. = 32 Since △ABC ~ △PQR, we have A = P i.e. x = 44° ∴ z = 50

  19. As As 410 33 + z = DEBC AEAC ADAB ADAB 410 2y = = = 3  104 3 + z= 2  104 y= Similar Triangles (p.189) • Example 2: In the figure, △ABC ~ △ADE. Find the unknowns. = 5 Since △ABC ~ △ADE, we have ACB = AED i.e. x = 104° 3 + z = 7.5 z = 4.5

  20. Time for Practice • Page 191 of Textbook 1B • Class Practice • Pages 191 – 192 of Textbook 1B • Questions 1 – 10 • Pages 80 – 83 of Workbook 1B • Questions 1 – 6

  21. Conditions for Triangles to be Similar (p.193) • There are three common conditions: • AAA: 3 Angles Equal • 3 sides prop.: 3 Sides Proportional • Ratio of 2 sides,: 2 Sides Proportional andinc.  their Included Angle Equal

  22. 3 Angles Equal (AAA, p.193) If A = X, B = Y and C = Z, then △ABC ~ △XYZ. [Reference: AAA]

  23. 3 Angles Equal (AAA, p.193) • Example 1: Are the two triangles in the figure similar? Give reasons. It is obvious that all corresponding angles are the same. Yes, △ABC ~ △LMN (AAA).

  24. 3 Angles Equal (AAA, p.193) • Example 2: In the figure, ADB and AEC arestraight lines. (a)Find ABC and ADE. (b)Write down a pair of similartriangles and givereasons. (a)In △ABC and △ADE, ABC ADE (b)△ABC ~ △AED (AAA) =180° – 60° –80° = 40° = 180° – 60° – 40° = 80°

  25. ax by cz 3 Sides Proportional (p.194) If = = , then △ABC ~ △XYZ. [Reference: 3 sides proportional]

  26. 3 Sides Proportional (p.194) • Example 1: Are the two triangles in the figure similar? Give reasons. It is noted that Yes, △LMN ~ △PQR (3 sides proportional).

  27. 3 Sides Proportional (p.194) • Example 2: Referring to the figure, writedown a pair of similartriangles and give reasons. It is noted that △ABC ~ △ACD (3 sides proportional)

  28. by cz 2 Sides Proportional and their Included Angle Equal (p.195) If = and a = x, then △ABC ~ △XYZ. [Reference: ratio of 2 sides, inc. ]

  29. 2 Sides Proportional and their Included Angle Equal (p.195) • Example 1: Are the two triangles in the figure similar? Give reasons. It is noted that Yes, △XYZ ~ △FED (ratio of 2 sides, inc.).

  30. 2 Sides Proportional and their Included Angle Equal (p.195) • Example 2: In the figure, ACE andBCD are straight lines. (a)Find DCE. (b)Write down a pair of similar triangles and give reasons. (a)DCE (b)△ABC ~ △EDC (ratio of 2 sides, inc.) (Why?) = ACB (Why?) = 54°

  31. ad be cf pr qs = , x = y = = Conditions for Triangles to be Similar (p.196) To conclude what we have learnt in this section, we can summarize the following conditions for two triangles to be similar. AAA 3 sides proportional ratio of 2 sides, inc. 

  32. Time for Practice • Page 198 of Textbook 1B • Class Practice • Pages 198 – 200 of Textbook 1B • Questions 1 – 10 • Pages 84 – 87 of Workbook 1B • Questions 1 – 6

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

  34. Enjoy the world of Mathematics! Ronald HUI

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