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CONGRUENCE AND SIMILARITY

The figures that have the same size and the same shape, i.e. one

shape fits exactly onto other is called Congruent figures .

CONGRUENT TRIANGLES:

1. Two triangles are congruent if they have the same size and the same shape.

2. If two triangles are congruent, then

a) Their corresponding angles are equal and

b) Their corresponding sides are equal.

3. If ABC = PQR, then

A P

B C Q R

1.1CONGRUENT FIGURES

A = P AB = PQ

B = Q BC = QR

C = R AC = PR

TIPS FOR STUDENTS: 1. The symbol means is congruent to 2. The matching diagram shows how the corresponding vertices match . ABC PQR, The corresponding vertices must be named in the correct order. e.g. BCA QRP and ACB PRQ

TEST OF CONGRUENCY BETWEEN TWO TRIANGLES:

TEST

DESCRIPTION

SSS

Two triangles are congruent if all Three Corresponding Sides are equal .

A P B C Q R This is known as the SSS rule. ( side, side, side )

If AB = PQ

BC = QR

AC = PR

ABC = PQR .

TEST

DESCRIPTION

SAS

Two triangles are congruent if Two Corresponding Sides and the Included angle are equal .

A P B C Q R This is known as the SAS rule. ( side, angle, side )

TEST

DESCRIPTION

AAS

Two triangles are congruent if Two Angles and a Corresponding Sides are equal . A P

B C Q R This is known as the AAS rule . ( angle, angle, side )

If AB = PQ

AC = PR and

A = P then

ABC = PQR .

If A = P and

B = Q then

BC = QR

ABC = PQR .

TEST

DESCRIPTION

ASA

Two triangles are congruent if Two angles and the Included Sides are equal .

A P B C Q R This is known as the ASA rule. ( angle, side, angle )

If A = P and

B = Q then

AB = PQ

ABC = PQR .

EXAMPLE1:

In the ABE CDE . BAE = 900, AED = 600, BC = 3 cm and DE = 18 cm.

FIND A

a) The length of AE ,

C

b) CDE . B E

3 600

18

D

TEST DESCRIPTION

RHS

Two triangles are congruent if both triangles have a Right Angle, equal Hypotenuse and another Side which is equal. A P B C Q R This is known as the RHS rule. ( right angle, hypotenuse, side )

If C = R

AB = PQ and

AC = PR then

ABC = PQR .

SOLUTION :

A

a) ABE = CDE ( Given )

BE = DE = 18 cm C 600

B E

3 15 300

CE = 18 - 3 cm = 15 cm

AE = CE = 15 cm 18

D

b) DCE = BAE = 900

CED = 600 2

= 300

CED = 1800 - 900 300 ( sum of )

EXAMPLE2:

In the quadrilateral ABCD, BE = CE and AE = DE .

Corresponding sides are equal .

Corresponding sides are equal .

Corresponding sides are equal .

AEB = CED sinceCorresponding sides are equal .

a) Prove that triangle AEB is congruent to B C

triangle DEC .

b) Name a triangle that is congruent to

triangle ABD .

c) Name a triangle that is congruent to

triangle ABC . A D

SOLUTION:

a) BE = CE

Given

AE = DE

AEB = DEC ( vert. opp. s )

AEB = DEC ( SAS rule )

b) DEC

E

BD = CA

BDA = CAD ( Base s of isos . AEB )

AD ( common sides )

AEB = DEC ( SAS rule )

c) DCB

1. Under a transformation, an object is formed onto its image .

2. A figure and its image are congruent under a translation, a rotation and a reflection .

Translation Rotation Reflection

Two figures are similar if

AC = DB

ACB = DBC ( Base s of isos . EBC )

BC ( common sides )

ABC = DCB ( SAS rule )

1.2 CONGRUENCE AND TRANSFORMATION

1.3 SIMILAR FIGURES

a)The corresponding angles are equal and

b)The corresponding sides are in the same ratio .

Two similar figures have the same shape but not necessarily the same size.

TIP FOR STUDENTS:

When two figures are congruent, they are also similar . However the converse is not true.

SIMILAR TRIANGLES:

1. Two triangles are similar if

a) Their corresponding angles are equal and

b) Their corresponding sides are in the same ratio.

2. If ABC is similar to PQR, then

A = P

B = Q and

=

=

C = R

P

A

B C Q R

TEST FOR SIMILARITY BETWEEN TWO TRIANGLES:

One of the following conditions is sufficient for two triangles to be similar.

1. Two triangles are similar if two of their corresponding angles are equal.

A P

B C

Q R

If A = P and

B = Q then

ABC