2. Congruence of Triangles
• Congruent triangles are triangles that have the
same size and shape. This means that the
corresponding sides are equal and the corresponding
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.
3. 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
4. ASA congruence rule
Two triangles are congruent if two angles and the
included side of one triangle are equal to two angles
and the included side of other triangle
Proof : We are given two triangles ABC and DEF in
which: ∠ B = ∠ E, ∠ C = ∠ F and BC = EF
To prove that : Δ ABC ≅ Δ DEF,
For proving the congruence of the two triangles see
that three cases arise.
5. ASA congruence rule
Case (i) : Let AB = DE in figure
You may observe that
AB = DE ……….(Assumed)
∠ B = ∠ E …………(Given)
BC = EF …………..(Given)
So, Δ ABC ≅ Δ DEF
…………….(By SAS rule)
A
B C
D
E Fl
l
6. A
B C
D
E Fll
Case (ii) : Let if possible AB > DE.
So, we can take a point P on AB
such that PB = DE.
Now consider Δ PBC and Δ DEF (see Fig.)
ASA congruence rule
P
In Δ PBC and Δ DEF,
PB = DE ……………………………(By construction)
∠ B = ∠ E ,BC = EF……………….. (Given)
So, Δ PBC ≅ Δ DEF, by the SAS axiom for congruence.
7. A
B C
D
E Fll
P
ASA congruence rule
Since the triangles are congruent, their
corresponding 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 = ED
So, Δ ABC ≅ Δ DEF …………………..(by SAS axiom)
8. Case (iii) : If AB < DE,
we can choose a point M on DE such that ME = AB
Δ ABC and Δ MEF (see Fig.)
AB = ME ……………………………(By construction)
∠ B = ∠ E ,BC = EF……………….. (Given)
So, Δ ABC ≅ Δ MEF, by the SAS axiom for congruence.
A
B C
D
E Fll
M
ASA congruence rule
9. If Δ ABC ≅ Δ 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, Δ ABC ≅ Δ DEF …………………..(by SAS axiom)
ASA congruence rule
A
B C
D
E Fll
M
10. • So all the three cases:-
• Case (i) : AB = DE
• Case (ii) : AB > DE
• Case (iii) : AB < DE,
We can see that Δ ABC ≅ Δ DEF
Proved
ASA congruence rule
A
B C
D
E Fll
11. SSS congruence rule
Two triangles are congruent, if three sides of one
triangle are equal to the corresponding three sides of
the other triangle
Given: Two Δ ABC and Δ DEF such that, AB = DE,
BC = EF, and AC = DF.
To Prove: To prove Δ ABC is congruent to Δ DEF.
A
B C
D
E F
12. SSS congruence rule
A
B C
D
E F
G
Construction: Let BC is the longest side.
Draw EG such that, < FEG = < ABC,
EG = AB. Join GF and GD
Proof: In Δ ABC & Δ GEF
BC = EF ……………….(Given)
AB = GE …………..(construction)
< ABC = < FEG ……(Construction)
Δ ABC ≅ Δ GEF
< BAC = < EGF and AC = GF
Now, AB = DE and AB = GE
DE = GE……..
Similarly,
AC = DF and AC = GF,
DF = GF
In Δ EGD, we have
DE = GE
< EDG = < EGD ---------- (i)
13. A
B C
D
E F
G
SSS congruence rule
In Δ FGD, we have
DF = GF……….(ii)
From (i) and (ii) we get,
< EDF = < EGF
But, <EGF = <BAC,
Therefore,
< EDF = < BAC -----------(iii)
In Δ ABC and Δ DEF, we have,
AB = DE ………..(Given)
< BAC = < EDG
AC = DF ……..(Given)
Therefore, by SAS congruence,
Δ ABC ≅ Δ DEF
14. If in two right triangles the hypotenuse and one side
of one triangle are equal to the hypotenuse and one
side of the other triangle, then the two triangles are
congruent.
Given: Two right angle Triangle
ABC and PQR where
AB = PQ and AC = PR,
To Prove:
Δ ABC ≅ Δ PQR
Proof:
we Know that these Triangles are Right angle
So < B = <C ……………..(900)
AB = PQ and AC = PR ------------(Given)
Δ ABC ≅ Δ PQR -------------(by SAS)
RHS congruence rule
A
BC
P
RQ
15. • Angles opposite to equal sides of an isosceles
triangle are equal.
Given : An isosceles Δ ABC in which
AB = 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 Δ BAD and Δ CAD,
AB = AC …………………..(Given)
∠ BAD = ∠ CAD …………(By construction)
Properties of a Triangle
A
B C
16. AD = AD ………………..(Common)
So, Δ BAD ≅ Δ 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
17. Call us for more information
www.iTutor.com
1-855-694-8886
Visit
