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

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

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

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

17. 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: A
B (ref. angle) C
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. 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: A B
C (ref. angle)
hypotenuse
opposite
adjacent

23. 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. W X Y

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

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

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

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

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

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

30. E.g. The figures X and Y as shown are congruent.
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. X Y
2. If two figures are congruent, then they will fit exactly on each other.

31. Therefore, there are two 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.
The line l divides the figure into 2 congruent figures, i.e and are congruent figures.
Therefore, there are two congruent figures.

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

33. Transformation and Congruence B)
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.

34. 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".

35. (ii) whether the two figures are congruent or not.
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.
(a)
____________
Reflection
Yes
Index

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

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

38. E.g. In the figure, if △ABC  △XYZ,
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, C A B Z X Y
then
∠A = ∠X,
∠B = ∠Y,
∠C = ∠Z,
AB = XY, BC = YZ,
CA = ZX.
and

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

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

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

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

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

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

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

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

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

48. 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)

49. Two Angles and One Side Equal C)
Conditions for Triangles to be Congruent
Example
Two Angles and One Side Equal C)
1. If ∠A = ∠X, AB = XY and ∠B = ∠Y, then △ABC  △XYZ.
【Reference: ASA】 C A B Z X Y

50. Two Angles and One Side Equal C)
Conditions for Triangles to be Congruent
Example
Two Angles and One Side Equal C)
2. If ∠A = ∠X, ∠B = ∠Y and BC = YZ, then △ABC  △XYZ.
【Reference: AAS】 C A B Z X Y

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

52. Conditions for Triangles to be Congruent
In the figure, equal angles are indicated with the same markings. Write down a pair of congruent triangles, and give reasons.
△ABD  △ACD (ASA)

53. Determine which pair(s) of triangles in the following are congruent.
Conditions for Triangles to be Congruent
1B_Ch11(53)
Determine which pair(s) of triangles in the following are congruent.
(I)
(II)
(III)
(IV)
In the figure, because of AAS,
(I) and (II) are a pair of congruent triangles;
(III) and (IV) are another pair of congruent triangles.

54. Conditions for Triangles to be Congruent
1B_Ch11(54)
In the figure, equal angles are indicated with the same markings. Write down a pair of congruent triangles, and give reasons. A B C D
△ABD  △CBD (AAS)

55. ‧ If ∠C = ∠Z = 90°, AB = XY and BC = YZ, then △ABC  △XYZ.
Conditions for Triangles to be Congruent
1B_Ch11(55)
Example
Two Right-angled Triangles with Equal Hypotenuses and Another Pair of Equal Sides D)
‧ If ∠C = ∠Z = 90°, AB = XY and BC = YZ, then △ABC  △XYZ.
【Reference: RHS】 A B C X Y Z

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

57. Conditions for Triangles to be Congruent
In the figure, ∠DAB and ∠BCD are both right angles and AD = BC. Judge whether △ABD and △CDB are congruent, and give reasons.
Yes, △ABD  △CDB (RHS)

58. The Meaning of Similarity
Similar Figures B
Similar Triangles

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

60. 2. Two congruent figures must be also similar figures.
The Meaning of Similarity
Example
Similar Figures A)
1. Two figures having the same shape are called similar figures. The figures A and B as shown is an example of similar figures.
2. Two congruent figures must be also similar figures.
3. When a figure is enlarged or reduced, the new figure is similar to the original one.

61. Find out all the figures similar to figure A by inspection.
The Meaning of Similarity
Find out all the figures similar to figure A by inspection. A B C D E
D, E

62. 1. If two triangles are similar, then
The Meaning of Similarity
Example
Similar Triangles B)
1. If two triangles are similar, then
i. their corresponding angles are equal;
ii. their corresponding sides are proportional.
In the figure, if △ABC ~ △XYZ, then ∠A = ∠X, ∠B = ∠Y,
∠C = ∠Z and A B C X Y Z

63. In the figure, given that △ABC ~ △PQR, find the unknowns x, y and z.
The Meaning of Similarity
In the figure, given that △ABC ~ △PQR, find the unknowns x, y and z.
x = 30° ,
y = 98° ∴ = ∴ =
z =
= 7.5

64. In the figure, △ABC ~ △RPQ. Find the values of the unknowns.
The Meaning of Similarity
In the figure, △ABC ~ △RPQ. Find the values of the unknowns.
Since △ABC ~ △RPQ,
∠B = ∠P ∴
x = 90°

65. Also, = Also, = = = = y z = ∴ y = 36 ∴ z = 65
The Meaning of Similarity
Back to Question
Also, =
Also, = = =
= y
z = ∴
y = 36 ∴
z = 65

66. Conditions for Triangles to be Similar
Example
Three Angles Equal A)
‧ If two triangles have three pairs of equal corresponding angles, then they must be similar.
【Reference: AAA】

67. Show that △ABC and △PQR in the figure are similar.
Conditions for Triangles to be Similar
Show that △ABC and △PQR in the figure are similar.
In △ABC and △PQR as shown,
∠B = ∠Q, ∠C = ∠R, ∠A
= 180° – 35° – 75°
= 70°
∠sum of  ∠P
= 180° – 35° – 75°
= 70°
∴ ∠A = ∠P
∴ △ABC ~ △PQR (AAA)

68. Are the two triangles in the figure similar? Give reasons.
Conditions for Triangles to be Similar
Are the two triangles in the figure similar? Give reasons.
【In △ABC, ∠B = 180° – 65° – 45° = 70°
In △PQR, ∠R = 180° – 65° – 70° = 45°】
Yes, △ABC ~ △PQR (AAA).

69. Three Sides Proportional B)
Conditions for Triangles to be Similar
Example
Three Sides Proportional B)
‧ If the three pairs of sides of two triangles are proportional, then the two triangles must be similar.
【Reference: 3 sides proportional】 a b c d e f

70. Show that △PQR and △LMN in the figure are similar.
Conditions for Triangles to be Similar
Show that △PQR and △LMN in the figure are similar.
In △PQR and △LMN as shown, , , ∴
∴ △PQR ~ △LMN (3 sides proportional)

71. Are the two triangles in the figure similar? Give reasons.
Conditions for Triangles to be Similar
Are the two triangles in the figure similar? Give reasons. 【 , 】
Yes, △ABC ~ △XZY (3 sides proportional).

72. Two Sides Proportional and their Included Angle Equal C)
Conditions for Triangles to be Similar
Example
Two Sides Proportional and their Included Angle Equal C)
‧ If two pairs of sides of two triangles are proportional and their included angles are equal, then the two triangles are similar.
【Reference: ratio of 2 sides, inc.∠】 x y p q r s

73. Show that △ABC and △FED in the figure are similar.
Conditions for Triangles to be Similar
Show that △ABC and △FED in the figure are similar.
In △ABC and △FED as shown,
∠B = ∠E , ∴
∴ △ABC ~ △FED (ratio of 2 sides, inc.∠)

74. Are the two triangles in the figure similar? Give reasons.
Conditions for Triangles to be Similar
Are the two triangles in the figure similar? Give reasons.
【 ∠ZYX = 180° – 78° – 40° = 62°, ∠ZYX = ∠CBA = 62°, , 】
Yes, △ABC ~ △XYZ (ratio of 2 sides, inc.∠).

75. Assignment
Perform Investigation 1 on page

76. Homework:
CYU # 6-11 on pages
Bring protractor next meeting!

77. Vectors and Bearings

78. WHAT IS A VECTOR? It describes the motion of an object.
A Vector comprises of
Direction
Magnitude (Size)
We will consider :
Column Vectors

79. Column Vectors a
Vector a
2 up
4 RIGHT
(4, 2)

80. Column Vectors
Vector b b
( -3, 2 )
2 up
3 LEFT
COLUMN Vector?

81. Column Vectors
Vector u n
( -4, -2 )
2 down
4 LEFT
COLUMN Vector?

82. Describing Vectors a b c d

83. Alternative LabellingF B D E G C A H

84. Generalization Vectors has both LENGTH and DIRECTION.

85. It is the angle of direction clockwise from north.
What is BEARINGS?
It is the angle of direction clockwise from north.

86. Bearings

87. Ex. The bearing of R from P is 220 and R is due west of Q
Ex. The bearing of R from P is 220 and R is due west of Q. Mark the position of R on the diagram.
Solution: P x Q x

88. Ex. The bearing of R from P is 220 and R is due west of Q
Ex. The bearing of R from P is 220 and R is due west of Q. Mark the position of R on the diagram.
Solution: P x . Q x

89. Ex. The bearing of R from P is 220 and R is due west of Q
Ex. The bearing of R from P is 220 and R is due west of Q. Mark the position of R on the diagram.
Solution: P
220 x . Q x
If you only have a semicircular protractor, you need to
subtract 180 from 220 and measure from south.

90. Ex. The bearing of R from P is 220 and R is due west of Q
Ex. The bearing of R from P is 220 and R is due west of Q. Mark the position of R on the diagram.
Solution: P x
40 . Q x
If you only have a semicircular protractor, you need to
subtract 180 from 220 and measure from south.

91. Ex. The bearing of R from P is 220 and R is due west of Q
Ex. The bearing of R from P is 220 and R is due west of Q. Mark the position of R on the diagram.
Solution: P
220 x . R Q x

92. Classwork 1) Answer the hand out given by the teacher.
2) Submit your work at the end of the class.
This will be evaluated. Make sure your scale drawing
is accurate and precise.

93. Homework: 1) Research about Pythagorean Theorem and its proof.
2) Answer the following questions:
Check Your Understanding
# 14, 15, 16 on pages 218.

94. Vectors and Bearings