Theorem 7.4

Transcript

Theorem 7.4 (SSS congruence rule) If three sides of a triangle are equal to the three sides of another triangle, then the two triangles are congruent) Given :- Δ PQR & Δ XYZ such that PQ = XY , QR = YZ , PR = XZ To Prove :- ∆PQR ≅ ∆XYZ Construction:- Draw XW intersecting YZ such that ∠WYZ = ∠PQR and WY = PQ. Also, Join WZ Proof:- In ∆PQR and ∆WYZ PQ = WY ∠PQR = ∠WYZ QR = YZ ∆PQR ≅ ∆WYZ Thus, ∠W = ∠P In Δ XYW Since PQ = WY and PQ = XY ∴ WY = XY ⇒ ∠1 = ∠2 Similarly, we can prove ∠3 = ∠4 Adding (2) and (3) we get ∠1 + ∠3 = ∠2 + ∠4 ⇒ ∠X = ∠W ⇒ ∠X = ∠W From (1), ∠W = ∠P ∴ ∠P = ∠X Now in ∆PQR and ∆XYZ PQ = XY ∠P = ∠X PR = XZ ⇒ ∆PQR ≅ ∆WYZ Hence Proved.