Math 10 Ms. Albarico

A.Modeling Situations Involving Right Triangles B.Congruence and Similarity 5.1 Ratios Based on Right Triangles

1)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. Students are expected to:

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

Introduction

Triangles Around Us

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

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

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

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

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...

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

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

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

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 opposite adjacent

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?

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

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

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

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

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

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

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

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

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

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

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

The Meaning of Similarity A Similar Figures B Similar Triangles

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

Congruent Figures 1.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. The Meaning of Congruence + Example Example A) 2.If two figures are congruent, then they will fit exactly on each other. XY

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

Find out by inspection the congruent figures among the following. The Meaning of Congruence ABCD EFGH B, D ;C, F

Transformation and Congruence ‧ 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. The Meaning of Congruence + Example Example B)

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".

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

(b) (i)____________ (ii)____________ (c) (i)____________ (ii)____________ Index The Meaning of Congruence Translation Yes Enlargement No + Back to QuestionBack to Question

The Meaning of Congruence Rotation Yes Reduction No + Back to QuestionBack to Question (d) (i)____________ (ii)____________ (e) (i)____________ (ii)____________

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

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

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

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

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

Determine which pair(s) of triangles in the following are congruent. Conditions for Triangles to be 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.

Each of the following pairs of triangles are congruent. Which of them are congruent because of SSS? Conditions for Triangles to be Congruent A B B

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

Determine which pair(s) of triangles in the following are congruent. Conditions for Triangles to be 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. (I)(II)(III)(IV)

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. Conditions for Triangles to be Congruent (a)(b) (a) △ ABC  △ CDA (SSS) (b) △ ACB  △ ECD (SAS)

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

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

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

In the figure, equal angles are indicated with the same markings. Write down a pair of congruent triangles, and give reasons. Conditions for Triangles to be Congruent △ ABD  △ ACD (ASA) Fulfill Exercise Objective  Identify congruent triangles from given diagram and given reasons.

Determine which pair(s) of triangles in the following are congruent. 1B_Ch11(55) Conditions for Triangles to be Congruent In the figure, because of AAS, (I) and (II) are a pair of congruent triangles; (III) and (IV) are another pair of congruent triangles. (I)(II)(III)(IV)

In the figure, equal angles are indicated with the same markings. Write down a pair of congruent triangles, and give reasons. 1B_Ch11(56) Conditions for Triangles to be Congruent A B C D △ ABD  △ CBD (AAS)

Two Right-angled Triangles with Equal Hypotenuses and Another Pair of Equal Sides Conditions for Triangles to be Congruent 1B_Ch11(57) + Example Example D) ‧ If ∠ C = ∠ Z = 90°, AB = XY and BC = YZ, then △ ABC  △ XYZ. 【 Reference: RHS 】 A BC X YZ

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

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

Similar Figures The Meaning of Similarity + Example Example 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.

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

Similar Triangles + Example Example B) 1.If two triangles are similar, then i.their corresponding angles are equal; ii.their corresponding sides are proportional. The Meaning of Similarity 2.In the figure, if △ ABC ~ △ XYZ, then ∠ A = ∠ X, ∠ B = ∠ Y, ∠ C = ∠ Z and. A B C X Y Z

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

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

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

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

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

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

Three Sides Proportional Conditions for Triangles to be Similar + Example Example 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

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

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

Two Sides Proportional and their Included Angle Equal Conditions for Triangles to be Similar + Example Example 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

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

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

Assignment Perform Investigation 1 on page

Homework: CYU # 6-11 on pages

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

a Vector a 4 RIGHT 2 up NOTE! Label is in BOLD. When handwritten, draw a wavy line under the label i.e. (4, 2)

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

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

Describing Vectors b a c d

Alternative Labelling A B C D F E G H

Generalization Vectors has both LENGTH and DIRECTION.

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

Bearings

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

P x. Q x Solution:

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

P x 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: Q x If you only have a semicircular protractor, you need to subtract 180 from 220 and measure from south. 40 .

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

Practice Exercises – Class work

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

Vectors and Bearings