can you give a brief proof for 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)
Side-Side-Side (SSS) Rule:
Side-Side-Side is a rule used to prove whether a given set of triangles are congruent.
The SSS rule states that
If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.
In the diagrams below, if AB = RP, BC = PQ and CA = QR, then triangle ABC is congruent to triangle RPQ.
It is perhaps the easiest to identify congruent triangles by using SSS congruence rule. The following video explains the application of SSS congruence rule.
ABCD is a rectangle with AC as one of its diagonals. Prove that the two triangles formed on either side of diagonal AC are congruent.
The required rectangle ABCD with AC as its diagonal can be drawn as
In ΔABC and ΔACD,
AB = CD (Opposite sides of a rectangle are equal)
BC = AD (Opposite sides of a rectangle are equal)
AC = AC (Common side)
Therefore, by SSS congruence rule, ΔABC ≅ ΔCDA
Thus, the two triangles formed on either side of diagonal AC are congruent.