TRIGONOMETRY

1. TRIGONOMETRY Math 10 Ms. Albarico

2. 5.1 Ratios Based on Right Triangles • Modeling Situations Involving Right Triangles • Congruence and Similarity

3. Students are expected to: Apply the properties of similar triangles. 2) Solve problems involving similar triangles and right triangles. 3) Determine the accuracy and precision of a measurement. 4) Solve problems involving measurement using bearings vectors.

4. Vocabulary perpendicular parallel sides angle triangle congruent similar dilate sail navigate approach

5. Introduction Trigonometry is a branch of mathematics that uses triangles to help you solve problems. Trigonometry is useful to surveyors, engineers, navigators, and machinists (and others too.)

6. Triangles Around Us

7. Review • Types of Triangles (Sides) • a) Scalene • b) Isosceles • c) Equilateral

9. Review • Types of Triangles (Angles) • a) Acute • b) Right • c) Obtuse

11. Labeling Right Triangles • The most important skill you need right now is the ability to correctly label the sides of a right triangle. • The names of the sides are: • the hypotenuse • the opposite side • the adjacent side

12. Labeling Right Triangles • The hypotenuse is easy to locate because it is always found across from the right angle. Since this side is across from the right angle, this must be the hypotenuse. Here is the right angle...

13. B C A Labeling Right Triangles • Before you label the other two sides you must have a reference angleselected. • It can be either of the two acute angles. • In the triangle below, let’s pick angle B as the reference angle. This will be our reference angle...

14. B (ref. angle) C A Labeling Right Triangles • Remember, angle B is our reference angle. • The hypotenuse is side BC because it is across from the right angle. hypotenuse

15. B (ref. angle) C A Labeling Right Triangles • Side AC is across from our reference angle B. So it is labeled: opposite. hypotenuse opposite

16. B (ref. angle) C A Labeling Right Triangles Adjacent means beside or next to • The only side unnamed is side AB. This must be the adjacent side. adjacent hypotenuse opposite

17. B (ref. angle) C A Labeling Right Triangles • Let’s put it all together. • Given that angle B is the reference angle, here is how you must label the triangle: hypotenuse adjacent opposite

18. Labeling Right Triangles • Given the same triangle, how would the sides be labeled if angle C were the reference angle? • Will there be any difference?

19. Labeling Right Triangles • Angle C is now the reference angle. • Side BC is still the hypotenuse since it is across from the right angle. B hypotenuse C (ref. angle) A

20. Labeling Right Triangles • However, side AB is now the side opposite since it is across from angle C. B opposite hypotenuse C (ref. angle) A

21. Labeling Right Triangles • That leaves side AC to be labeled as the adjacent side. B hypotenuse opposite C (ref. angle) A adjacent

22. B hypotenuse opposite C (ref. angle) A adjacent Labeling Right Triangles • Let’s put it all together. • Given that angle C is the reference angle, here is how you must label the triangle:

23. W X Y Labeling Practice • Given that angle X is the reference angle, label all three sides of triangle WXY. • Do this on your own. Click to see the answers when you are ready.

24. W X Y Labeling Practice • How did you do? • Click to try another one... adjacent opposite hypotenuse

25. R T S Labeling Practice • Given that angle R is the reference angle, label the triangle’s sides. • Click to see the correct answers.

26. R T S Labeling Practice • The answers are shown below: hypotenuse adjacent opposite

27. Which side will never be the reference angle? The right angle What are the labels? Hypotenuse, opposite, and adjacent

28. A B C Congruent Figures Transformation and Congruence Congruent Triangles The Meaning of Congruence

29. Three Sides Equal A B C D Two Sides and Their Included Angle Equal Two Angles and One Side Equal Two Right-angled Triangles with Equal Hypotenuses and Another Pair of Equal Sides Conditions for Triangles to be Congruent

30. A B Similar Figures Similar Triangles The Meaning of Similarity

31. A B C Three Angles Equal Three Sides Proportional Two Sides Proportional and their Included Angle Equal Conditions for Triangles to be Similar

32. X Y The Meaning of Congruence • Example Congruent Figures A) • Two figures having the same shape and the same size are called congruent figures. • E.g. The figures X and Y as shown are congruent. 2. If two figures are congruent, then they will fit exactly on each other.

33. The line l divides the figure into 2 congruent figures,i.e. and are congruent figures. The Meaning of Congruence The figure on the right shows a symmetric figure with l being the axis of symmetry. Find out if there are any congruent figures. Therefore, there are two congruent figures.

34. A B C D E F G H The Meaning of Congruence Find out by inspection the congruent figures among the following. B, D ; C, F

35. The Meaning of Congruence • Example Transformation and Congruence B) ‧ When a figure is translated, rotated or reflected, the image produced is congruent to the original figure. When a figure is enlarged or reduced, the image produced will NOT be congruent to the original one.

36. Note: A DILATATION is a transformation which enlarges or reduces a shape but does not change its proportions. SIMILARITY is the result of dilatation "≅" means "is congruent to " "~" is "similar to".

37. (a) • ____________ • ____________ The Meaning of Congruence In each of the following pairs of figures, the red one is obtained by transforming the blue one about the fixed point x. Determine (i) which type of transformation (translation, rotation, reflection, enlargement, reduction) it is, (ii) whether the two figures are congruent or not. Reflection Yes Index

38. (b) • ____________ • ____________ (c) • ____________ • ____________ The Meaning of Congruence • Back to Question Translation Yes Enlargement No Index

39. (d) • ____________ • ____________ (e) • ____________ • ____________ The Meaning of Congruence • Back to Question Rotation Yes Reduction No

40. C Z A X B Y The Meaning of Congruence • Example Congruent Triangles C) ‧When two triangles are congruent, all their corresponding sides and corresponding angles are equal. E.g. In the figure, if △ABC △XYZ, then ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z, AB = XY,BC = YZ, CA = ZX. and

41. Trigonometry Name a pair of congruent triangles in the figure. From the figure, we see that △ABC △RQP.

42. For two congruent triangles, their corresponding sides and angles are equal. ∴ p = 6 cm , q = 5 cm , r = 50° ∴ Trigonometry Given that △ABC△XYZ in the figure, find the unknowns p, q and r.

43. T A (a) (b) Y Z P U S C B Q X R Trigonometry Write down the congruent triangles in each of the following. (a) △ABC△XYZ (b) △PQR△STU

44. P X A 98° 15 35° N z 47° C 13 M 14 Y B x Z Trigonometry Find the unknowns (denoted by small letters) in each of the following. (a) △ABC△XYZ (b) △MNP△IJK I i K j J (a) x = 14 , z = 13 (b) j = 35°, i = 47°

45. ‧ If AB = XY, BC = YZ and CA = ZX, then △ABC△XYZ. 【Reference: SSS】 X Z Y A C B Conditions for Triangles to be Congruent • Example Three Sides Equal A)

46. (I) (II) (III) (IV) Conditions for Triangles to be Congruent Determine which pair(s) of triangles in the following are congruent. In the figure, because of SSS, (I) and (IV) are a pair of congruent triangles; (II) and (III) are another pair of congruent triangles.

47. 5 A B 10 3 7 3 18 5 18 7 10 Conditions for Triangles to be Congruent Each of the following pairs of triangles are congruent. Which of them are congruent because of SSS? B

48. ‧ If AB = XY, ∠B = ∠Y and BC = YZ, then △ABC△XYZ. 【Reference: SAS】 A X C Z B Y Conditions for Triangles to be Congruent • Example Two Sides and Their Included Angle Equal B)

49. (I) (II) (III) (IV) Conditions for Triangles to be Congruent Determine which pair(s) of triangles in the following are congruent. In the figure, because of SAS, (I) and (III) are a pair of congruent triangles; (II) and (IV) are another pair of congruent triangles.

50. Conditions for Triangles to be Congruent In each of the following figures, equal sides and equal angles are indicated with the same markings. Write down a pair of congruent triangles, and give reasons. (a) (b) (a) △ABC△CDA (SSS) (b) △ACB△ECD (SAS)