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Chapter 10Congruent and Similar Triangles

Introduction

Recognizing and using congruent and similar

shapes can make calculations and design work

easier. For instance, in the design at the

corner, only two different shapes were actually

drawn. The design was put together by copying and

manipulating these shapes to produce versions of

them of different sizes and in different

positions.

In this chapter, we will look in a little more

depth at the mathematical meaning of the terms

similar and congruent, which describe the

relation between shapes like those in design.

Similar and Congruent Figures

- Congruent polygons have all sides congruent and

all angles congruent. - Similar polygons have the same shape they may or

may not have the same size.

Worksheet Exercise 1 Which of the following

pairs are congruent and which are similar?

Examples

These figures are similar and congruent. Theyre

the same shape and size.

These figures are similar but not congruent.

Theyre the same shape, but not the same size.

Another Example

- These figures are neither similar nor congruent.

Theyre not the same shape or the same size.

Even though theyre both triangles, theyre not

similar because theyre not the same shape

triangle.

Note Two figures can be similar but not

congruent, but they cant be congruent but not

similar. Think about why!

Congruent Figures

- When 2 figures are congruent, i.e. 2 figures have

the same shape and size, - Corresponding angles are equal
- Corresponding sides are equal
- Symbol ?

Congruent Triangles

- AB XY, BC YZ, CA ZX
- ?A ?X , ?B ?Y, ?C ?Z

Note Corresponding vertices are named in order.

THE ANGLE MEASURES OF A TRIANGLE AND CONGRUENT

TRIANGLES

- The sum of the angle measures of a triangle is

180o

Example

? 85o

- Congruent triangles

Congruent triangles are triangles with the same

shape and size

Angle 60o side 5cm

Example

Isosceles triangles

- An isosceles triangle is the triangle which has

at least two sides with the same length

- In an isosceles triangle, angles that

are opposite the equal-length sides have the same

measure

Example

The side 82 cm, the angle 76o

Equilateral triangles

- An equilateral triangle has three sides of equal

length

- In an equilateral triangle, the measure of each

angle is 60o

Example

Angle 60o, side 100 cm

Right triangles and Pythagorean theorem

- A right triangle is the triangle with one right

angle

- Pythagorean theorem
- c2 a2 b2

Example

c2 42 32 25

C 5

Ex 10A Page 47

- Q2 b
- By comparing,
- x 16,
- y 30
- ( 180?- 75?- 75?)

- Q2 a
- By comparing,
- x 4.8,
- y 42

- Q2 d
- By comparing,
- x 22,
- y 39 22
- 17

Tests for Congruency

- Ways to prove triangles congruent
- SSS ( Side Side Side )
- SAS ( Side Angle Side )
- ASA ( Angle Side Angle ) or AAS ( Angle

Angle Side ) - RHS ( Right angle Hypotenuse Side )

SSS ( Side Side Side )

- Three sides on one triangle are equal to three

sides on the other triangle.

- AB XY,
- BC YZ,
- CA ZX

(SSS)

Example

- Given AB DB and AC DC.
- Prove that ?ABC ? ?DBC
- AB DB ( Given )
- AC DC ( Given )
- BC ( common)
- Hence ?ABC ? ?DBC ( SSS )

Textbook Page 44 Ex 10A Q 1 a, k

SAS ( Side Angle Side )

- Two pairs of sides and the included angles are

equal.

- AB XY,
- BC YZ,
- ?ABC ?XYZ
- ( included angle )

(SAS)

Example

B

E

A

C

- Given AC EC and BC DC.
- Prove that ?ABC ? ?EDC
- AC EC ( Given )
- ?ACB ?ECD ( included angle, vert opp )
- BC DC ( Given )
- Hence ?ABC ? ?EDC ( SAS )

D

Textbook Page 44 Ex 10A Q 1 c, i

ASA ( Angle Side Angle )AAS ( Angle Angle

Side )

- Two pairs of angles are equal and a pair of

corresponding sides are equal.

- AB XY,
- ?ABC ?XYZ
- ?BAC ?YXZ

(ASA)

From given diagram, ?ACB ?XZY

(AAS)

Example

B

E

A

C

- Given AC EC and ?BAC ?DEC
- Prove that ?ABC ? ?DEC
- AC EC ( Given )
- ?BAC ?DEC ( Given )
- ?ACB ?ECD (vert opp)
- Hence ?ABC ? ?EDC ( ASA )

D

Textbook Page 44 Ex 10A Q 1 f, o

RHS ( Right angle Hypotenuse Side )

- Right-angled triangle with the hypotenuse equal

and one other pair of sides equal.

- ?ABC ?XYZ 90
- ( right angle)
- AC XZ ( Hypotenuse)
- BC YZ

(RHS)

Example

- Prove that ?ABC ? ?DBC
- ?ACB ?DCB 90?
- AB DB ( Given, hypotenuse )
- BC is common
- Hence ?ABC ? ?EBC ( RHS )

Textbook Page 44 Ex 10A Q 1 g, j Try Q1 e , 1y

too

Time to work

- Home Work
- Ex 10A Page 44-47
- Q 1 b, h, m, p, r, x
- Q 2 c, e
- Ex 10B Pg 49-50
- Q3, 5, 7, 8

- Class Work
- Ex 10B Pg 49
- Q1
- Q2
- Q4
- Q6

Thinking Time ?????

- If 3 angles on ?A are equal to the 3

corresponding angles on the other ?B, are the two

triangles congruent ?

Ratios and Similar Figures

- Similar figures have corresponding sides and

corresponding angles that are located at the same

place on the figures. - Corresponding sides have to have the same ratios

between the two figures.

Ratios and Similar Figures

A

B

E

F

Example

G

H

C

D

Ratios and Similar Figures

Example

These rectangles are similar, because the ratios

of these corresponding sides are equal

Proportions and Similar Figures

- A proportion is an equation that states
- that two ratios are equivalent.

- Examples

- n 5 m 4

Proportions and Similar Figures

You can use proportions of corresponding sides to

figure out unknown lengths of sides of polygons.

10/16 5/n so n 8 m

Similar triangles

- Similar triangles are triangles with the same

shape

For two similar triangles,

- corresponding angles have the same measure
- length of corresponding sides have the same ratio

Example

Side 6 cm

Angle 90o

Similar Triangles

- 3 Ways to Prove Triangles Similar

Similar triangles are like similar polygons.

Their corresponding angles are CONGRUENT and

their corresponding sides are PROPORTIONAL.

10

5

6

3

8

4

But you dont need ALL that information to be

able to tell that two triangles are similar.

AA Similarity

- If two angles of a triangle are congruent to the

two corresponding angles of another triangle,

then the triangles are similar.

25 degrees

25 degrees

SSS Similarity

- If all three sides of a triangle are proportional

to the corresponding sides of another triangle,

then the two triangles are similar.

21

14

18

8

12

12

SSS Similarity Theorem

If the sides of two triangles are in proportion,

then the triangles are similar.

D

A

B

C

E

F

SAS Similarity

- If two sides of a triangle are proportional to

two corresponding sides of another triangle AND

the angles between those sides are congruent,

then the triangles are similar.

14

21

18

12

SAS Similarity Theorem

D

A

B

C

E

F

If an angle of one triangle is congruent to an

angle of another triangle and the sides including

those angles are in proportion, then the

triangles are similar.

D

A

B

C

E

F

SAS Similarity Theorem

Idea for proof

Name Similar Triangles and Justify Your Answer!

A

80?

D

E

80?

B

C

?ABC ?ADE by AA Postulate

C

6

10

D

E

5

3

A

B

?CDE ?CAB by SAS Theorem

L

5

3

M

6

6

N

K

6

10

O

?KLM ?KON by SSS Theorem

A

20

D

30

24

16

B

C

36

?ACB ?DCA by SSS Theorem

L

15

P

A

25

9

N

?LNP ?ANL by SAS Theorem

Time to work !!!!

- Class work
- Ex 10C Page 54
- Q2a to h
- Q3
- Q5
- Q6 a to d
- Q8
- Q10
- Q12
- Q13

- Home work
- Ex 10C Page 54
- Q1a to f
- Q4
- Q7
- Q9
- Q11
- Q14
- Q15

Areas of Similar Figures

Activity Complete the table for each of the

given pairs of similar figures

Conclusion

If the ratio of the corresponding lengths of two

similar figures is

then the ratio of their areas is

Thinking Time

- Does the identity works for the following figures

? Why?

Time to work !!!

- Class work
- Ex 10D Pg 62
- Q 10
- Q12
- Q13
- Q15
- Q16
- Q20 - 22

- Class work
- Ex 10D Pg 62
- Q1 a to d
- Q3
- Q4
- Q5
- Q8
- Q9

Home Work ?

- Ex 10D Pg 62
- Q2
- Q6
- Q7
- Q11
- Q14
- Q17
- Q18

Volumes of Similar Solids

Activity Complete the table for each of the

given pairs of similar Solids

Conclusion

If the ratio of the corresponding lengths of two

similar figures is

then the ratio of their volumes is

Total Surface Area of similar solids

If the ratio of the corresponding lengths of two

similar figures is

then the ratio of their total surface areas is

Time to work ?

- Class work
- Ex 10E Pg 67
- Q1
- Q2
- Q3 ?
- Q4
- Q5

- Class work
- Ex 10E Pg 67
- Q6
- Q9
- Q11

Q3 ? How to find the weight of a similar solid???

- If both solids were made from the same material,
- Density will be the same ?
- Hence using the formula
- Density Mass ? Volume

Your favourite moment ?

- Home work
- Ex 10E Pg 67
- Q7
- Q8
- Q10
- Q12
- Q13
- Q14

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