Chapter 10Congruent and

Similar Triangles

IntroductionRecognizing 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?

ExamplesThese figures are similar and congruent. They’re the same shape and size.

These figures are similar but not congruent. They’re the same shape, but not the same size.

Another ExampleThese figures are neither similar nor congruent. They’re not the same shape or the same size. Even though they’re both triangles, they’re not similar because they’re not the same shape triangle.

Note: Two figures can be similar but not congruent, but they can’t be congruent but not similar. Think about why!

Congruent FiguresWhen 2 figures are congruent, i.e. 2 figures have the same shape and size,Corresponding angles are equalCorresponding sides are equalSymbol :

Congruent Triangles

• AB = XY, BC = YZ, CA = ZXA =X , B =Y, C =Z

ABC XYZ A

C

B

X

Z

Y

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

30o

65o

?

? = 85o

• Congruent triangles

90o

60o

5 cm

?

?

Example

Congruent triangles are triangles with the same shape and size

Angle = 60o; side = 5cm

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

The side = 82 cm, the angle = 76o

82cm

52o

?

?Example

Equilateral triangles

• An equilateral triangle has three sides of equal length

• In an equilateral triangle, the measure of each angle is 60o

Example 60o

100cm

?

?

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

Leg

a

Leg

b

Hypotenuse

c

c2 = 42 + 32 = 25

60o

?3 cm

4 cm

?

C = 5

Ex 10A Page 47• Q2 a

• By comparing,

x = 4.8,

y = 42

• Q2 d

• By comparing,

x = 22,

y = 39 – 22

= 17

• Q2 b

• By comparing,

x = 16,

y = 30

( 180- 75- 75)

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.A

B CX

Y

Z

•AB = XY,

•BC = YZ,

•CA = ZX

ABC XYZ (SSS)

Example :

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

A

C

B D

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 )

ABC XYZ (SAS)

A

B C

XY

Z

Example :

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 )

Textbook Page 44 Ex 10A Q 1 c, i

A

B

C

D

E

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 = XYZBAC = YXZ

ABC XYZ (ASA)

A

B CX

Y

Z

From given diagram, ACB = XZY ABC XYZ (AAS)

Example :

Given AC = EC and BAC = DECProve that ABC DEC• AC = EC ( Given )BAC = DEC ( Given )ACB = ECD (vert opp)• Hence ABC EDC ( ASA )

Textbook Page 44 Ex 10A Q 1 f, o

A

B

C

D

E

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

ABC XYZ (RHS)

A

B C

XY

Z

Example :

Prove that ABC DBCACB = 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

DA

B

C

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

Example

A E

C

F

D

G H

B

These sides correspond:

AB and EF

BD and FH

CD and GH

AC and EG

These angles correspond:

A and E

B and F

D and H

C and G

Ratios and Similar Figures

Example

7 m

3 m 6 m

14 m

These rectangles are similar, because the ratios of these corresponding sides are equal:

7 14

3 6

3 6

7 14

7 3

14 6

14 6

7 3

•A proportion is an equation that states

that two ratios are equivalent.

•Examples:

4 8

10n

6

3 2

m

n = 5 m = 4

Proportions and Similar Figures

Proportions and Similar Figures

You can use proportions of corresponding sides to figure out unknown lengths of sides of polygons. 16

m

10 m

n

5 m

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

65o

25o

?4 cm

2cm

12cm?

Example

Angle = 90o Side = 6 cm

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.

610

8

3

4

5

But you don’t 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.

18

12

8

12

1421

2

3

14

212

3

12

182

3

8

12

SSS Similarity Theorem

If the sides of two triangles are in proportion, then the triangles are similar.

A

BC

D

EF

EF

BC

DF

AC

DE

AB

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.

1821

12

14

A

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.

BC

D

EF

DF

AC

DE

AB

DA

SAS Similarity Theorem

SAS Similarity Theorem

D

EF

DF

AC

DE

AB

DA

A

BC

Idea for proof

Name Similar Triangles and Justify Your Answer!

A

B C

D E80

80

ABC ~ ADE by AA ~ Postulate

A B

C

D E

CDE~ CAB by SAS ~ Theorem

6

3

10

5

O

N

L

KM

KLM~ KON by SSS ~ Theorem

63

10

56

6

CB

A

D

ACB~ DCA by SSS ~ Theorem

24

36

20

3016

N

L

AP

LNP~ ANL by SAS ~ Theorem

25 9

15

Time to work !!!!oClass work

Ex 10C Page 54

Q2a to h

Q3

Q5

Q6 a to d

Q8

Q10

Q12

Q13

oHome work

Ex 10C Page 54

Q1a to f

Q4

Q7

Q9

Q11

Q14

Q15

Areas of Similar FiguresActivity : 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 a

bthen the ratio of their areas is

2a

b

2

1 1

2 2

A

A

l

l

Thinking Time

Does the identity works for the following figures ? Why?

Time to work !!!

2

1 1

2 2

A

A

l

l

• 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 SolidsActivity : 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 a

bthen the ratio of their volumes is

3a

b

3

1 1

2 2

V

V

l

l

Total Surface Area of similar solids

If the ratio of the corresponding lengths of two similar figures is a

b

then the ratio of their total surface areas is 2

a

b

2

1 1

2 2

SA

SA

l

l

Time to work

• Class work

• Ex 10E Pg 67

• Q1

• Q2

• Q3 • Q4

• Q5

2

1 1

2 2

SA

SA

l

l

3

1 1

2 2

V

V

l

l

• 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

1 1

2 2

3

1 1

2 2

V

V

M V

M V

l

l

1 1

2 2

and

M V

M V

3

1 1

2 2

M l

M l

Your favourite moment • Home work

• Ex 10E Pg 67

• Q7

• Q8

• Q10

• Q12

• Q13

• Q14