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: Î”PQR and Î”XYZ are such that PQ = XY, QR = YZ and PR = XZ To prove: Î”PQR Î”XYZ Construction: Draw YW such that âˆ ZYW = âˆ PQR and WY = PQ Join XW and WZ Proof: In Î”PQR and Î”WYZ QR = YZ (Given) âˆ PQR = âˆ ZYW (Construction) PQ = YW (Construction) âˆ´Î”PQR Î”WYZ (SAS congruence criterion) â‡’âˆ P =âˆ W and PR = WZ (CPCT) PQ = XY and PQ = XW âˆ´ XY = YW Similarly, XZ = WZ In Î”XYW, XY = YW â‡’ âˆ YWX = âˆ YXW (In a triangle, equal sides have equal angles opposite to them) Similarly, âˆ ZWX = âˆ ZXW âˆ´ âˆ YWX + âˆ ZWX = âˆ YXW + âˆ ZXW â‡’ âˆ W = âˆ X Now, âˆ W = âˆ P âˆ´ âˆ P = âˆ X In Î”PQR and Î”XYZ , PQ = XY âˆ P = âˆ X PR = XZ âˆ´ Î”PQR Î”XYZ (SAS congruence criterion)

sss congruence 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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