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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.
Proof: In BAD and CAD, AB = AC ..(Given)
BAD = CAD (By construction)
Properties of a Triangle
A
B C
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AD = AD ..(Common)So, BAD CAD (By SAS rule)
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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