PLEASE PROVE SSS RULE
 

SSS congruence criterion:

Two triangles are congruent if the three sides of one triangle are equal to the corresponding three sides of the other triangle

Given: ΔPQR and ΔXYZ are such that PQ = XY, QR = YZ and PR = XZ.

To prove: ΔPQR ΔXYZ

Construction: Draw YW such that ∠ZYW = ∠PQR and WY = PQ. Join XW and WZ.

Proof: In  ΔPQR and  ΔWYZ

QR = YZ    (Given)

∠PQR = ∠ZYW  (Construction)

PQ = YW  (Construction)

∴ΔPQR ΔWYZ     (SAS congruence criterion)

⇒∠P  =∠W and PR = WZ      (CPCT)

PQ = XY and PQ = XW

∴ XY = YW

Similarly, XZ = WZ

In ΔXYW, XY = YW

⇒ ∠YWX = ∠YXW         (In a triangle, equal sides have equal angles opposite to them)

Similarly, ∠ZWX = ∠ZXW

∴ ∠YWX + ∠ZWX = ∠YXW + ∠ZXW

⇒ ∠W = ∠X

Now, ∠W = ∠P

∴ ∠P = ∠X

In ΔPQR and ΔXYZ ,

PQ = XY

∠P = ∠X

PR = XZ

∴ ΔPQR ΔXYZ              (SAS congruence criterion)