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# The E Tutor-Congruence of the Triangle

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TRIANGLES

Congruence of Triangles

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CongruenceofTriangles Congruent triangles are triangles that have the

same size and shape. This means that thecorresponding sides are equal and thecorresponding angles are equal

In the above diagrams, the corresponding sides

are a and d; b and e ; c and f. The corresponding angles

are x and s; y and t; z and u.

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Criteria for Congruence of Triangles

There are four rules to check for congruent triangles. SSS Rule (Side-Side-Side rule) SAS Rule (Side-Angle-Side rule) ASA Rule (Angle-Side-Angle Rule)

AAS Rule (Angle-Angle-Side rule) Hypotenuse Leg Rule

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ASA congruence rule

Two triangles are congruent if two angles and the

included side of one triangle are equal to two anglesand the included side of other triangle

Proof : We are given two triangles ABC and DEF inwhich: B = E, C = F and BC = EF

To prove that : ABC DEF ,

For proving the congruence of the two triangles seethat three cases arise.

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ASA congruence rule

Case (i) : Let AB = DE in figure

You may observe that

AB = DE .(Assumed) B = E (Given)BC = EF ..(Given)So, ABC DEF

.(By SAS rule)

A

B C

D

E Fl

l

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A

B C

D

E Fl

l

Case (ii) : Let if possible AB > DE.So, we can take a point P on ABsuch that PB = DE.

Now consider PBC and DEF (see Fig.)

ASA congruence rule

P

In PBC and DEF, PB = DE (By construction)

B = E ,BC = EF.. (Given)So, PBC DEF, by the SAS axiom for congruence.

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A

B C

D

E Fl

l

P

ASA congruence rule

Since the triangles are congruent, theircorresponding parts will be equal.

So, PCB = DFE But, given that ACB = DFE

So, ACB = PCB This is possible only if P coincides with A.

or, BA = EDSo, ABC DEF ..(by SAS axiom)

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Case (iii) : IfAB < DE,

we can choose a point M on DE such that ME = AB

ABC and MEF (see Fig.)

AB = ME (By construction)

B = E ,BC = EF.. (Given)

So, ABC MEF, by the SAS axiom for congruence.

A

B C

D

E F

l

l

M

ASA congruence rule

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If ABC MEF

then corresponding parts will be equal.So, ACB = MFE, But ACB = DFE

(Given)

so, ACB = MCB

This is possible only if M coincides with D.or, BA = ED

So, ABC DEF ..(by SAS axiom)

ASA congruence rule

A

B C

D

E F

ll

M

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So all the three cases:-

Case (i) : AB = DE Case (ii) : AB > DE Case (iii) : AB < DE,

We can see that ABC DEF

Proved

ASA congruence rule

A

B C

D

E F

ll

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SSS congruence rule

Two triangles are congruent, if three sides of one

triangle are equal to the corresponding three sides ofthe other triangle

Given: Two ABC and DEF such that, AB = DE, BC = EF, and AC = DF.

To Prove:To prove ABC is congruent to DEF.

A

B C

D

E F

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SSS congruence rule

A

B CD

E F

G

Construction: Let BC is the longest side.Draw EG such that, < FEG = < ABC,

EG = AB. Join GF and GDProof: In ABC & GEF

BC = EF .(Given)AB = GE ..(construction)

< ABC = < FEG (Construction) ABC GEF < BAC = < EGF and AC = GF

Now, AB = DE and AB = GEDE = GE..

Similarly,AC = DF and AC = GF,

DF = GFIn EGD, we have

DE = GE

< EDG = < EGD ---------- (i)

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A

B CD

E F

G

SSS congruence ruleIn FGD, we have

DF = GF.(ii)

From (i) and (ii) we get,

< EDF = < EGF

But,

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If in two right triangles the hypotenuse and one sideof one triangle are equal to the hypotenuse and one

side of the other triangle, then the two triangles arecongruent.

Given:Two right angle Triangle

ABC and PQR where

AB = PQ and AC = PR,

To Prove:

ABC PQR

Proof: we Know that these Triangles are Right angle

So < B =

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Angles opposite to equal sides of an isoscelestriangle are equal.

Given:An isosceles ABC in whichAB = AC.

To prove: B = C

Construction:

Draw the bisector of A

and D be the point of intersection of this bisector of

A and BC.

Properties of a Triangle

A

B C

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So, ABD = ACD,

since they are

corresponding angles of congruent triangles.

So,

B = C

Proved

Properties of a Triangle

A

B C

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