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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)

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