Criteria for congruency of triangles

Suppose, we have two triangles

To prove ∆ABC ≅ ∆PQR,

We use different congruency criteria

They are

• SSS – Side Side Side
• SAS − Side Angle Side
• ASA – Angle Side Angle
• AAS − Same as ASA
• AAA – Angle Angle Angle (Check Why AAA is not a congruence )
• RHS − Right-angle Hypotenuse side

Let’s discuss them

SSS

In ∆ABC & ∆PQR

  AB = PQ                (Both are 5)

  BC = QR               (Both are 4)

  AC = PR               (Both are 3)

∴ ∆ABC ≅ ∆PQR   (SSS Congruence Rule)

For more details,

Please check SSS Explanation  & proof

Note :

Here ∆ABC ≅ ∆PQR

   not ∆ABC ≅ ∆QRP

Order of writing the triangle is important

SAS

In ∆ABC & ∆PQR

  AB = PQ                  ( Both are 5 )

  ∠B  = ∠Q                 ( Both are 60° )

  BC = QR                 ( Both are 4 )

∴ ∆ABC ≅ ∆PQR    ( SAS Congruence Rule )

For more details,

Please check SAS Explanation  & proof

ASA

In ∆ABC & ∆PQR

  ∠B = ∠Q         ( Both are 40 °)

  BC  = QR       ( Both are 5 cm )

  ∠C = ∠R        ( Both are 80 °) 

∴ ∆ABC ≅ ∆PQR

For more details,

Please check ASA Explanation & proof

RHS

In ∆ABC & ∆PQR

∠B = ∠Q      ( Right angle, both 90 °)

  AC  = PR    ( Hypotenuse, both 5 cm )

  AB = PQ     (Side , both 4 cm )

 ∴ ∆ABC ≅ ∆PQR        ( RHS Congruency rule )

For more details,

Please check RHS Explanation  & proof