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Criteria for Congruence of Triangles

Two triangles are congruent if 3 pairs of corresponding sides
and 3 pairs of corresponding angles are equal.
If DABC ≅ DDEF, then

AB = DE, BC = EF, CA = FD
and ∠A = ∠D, ∠B = ∠E, ∠C = ∠F.

For checking the congruence of two triangles, it is sufficient to check only 3 elements of the two
triangles. There are five cases for the two triangles to be congruent.

1. SSS (Side-Side-Side) 2. SAS (Side-Angle-Side) 3. ASA (Angle-Side-Angle)
4. AAS (Angle-Angle-Side) 5. RHS (Right Angle-Hypotenuse-Side)

We will discuss them one by one.
SSS Congruence Criterion (Side-Side-Side)

Two triangles are congruent, if three sides of one triangle are respectively equal to the three
sides of the other triangle.
Draw DABC in which AB = 5 cm, BC = 6 cm, CA = 7 cm.
Draw another DPQR with PQ = 5 cm, QR = 6 cm, RP = 7 cm.
Now make a trace copy of DPQR and place it on DABC in
such a way that PQ falls on AB and QR falls on BC.

You will observe that RP coincides with CA.
Two triangles cover each other exactly. Therefore, DABC ≅ DPQR.

Since, both the triangles cover each other exactly. Therefore their corresponding angles are also
equal. i.e., ∠A = ∠P, ∠B = ∠Q and ∠C = ∠R.
SAS Congruence Criterion (Side-Angle-Side)

Two triangles are congruent if two sides and the included angle of one triangle are respectively
equal to the two sides and the included angle of the other triangle.

Draw DABC and DDEF according to the given figure. A D
Thus, AB = DE, BC = EF and ∠B = ∠E.
3 cm 3 cm
Make a trace copy of DABC on a tracing paper and try
to cover DDEF with A on D, B on E and C on F. 65° 65°
B C E F
It is observed that two triangles cover each other 4 cm 4 cm
exactly. Therefore, DABC ≅ DDEF.

ASA Congruence Criterion (Angle-Side-Angle)

Two triangles are congruent if two angles and the included
side of one triangle are respectively equal to the two angles
and the included side of the other triangle.

Draw DABC with BC = 4 cm, ∠B = 60º and ∠C = 40º.

187 The Triangle and Its Properties