Converse of Pythagoras Theorem
Theorem :
In a triangle if the square of one side is equal to the sum of the squares of the remaining two sides, then the triangle is a right angled triangle.
Given :
In ∆ABC,
AC² = AB² + BC²
To prove :
∆ABC = 90°
Construction :
Draw ∆PQR such that, AB = PQ, BC = QR, ∆PQR = 90°.
Proof :
In ∆PQR,
∠Q = 90°
∴ PR² = PQ² + QR² …[By Pythagoras theorem]
∴ PR² = AB² + BC² …(i) [By construction]
∴ PR² = AC² …(ii) [Given]
∴ PR = AC …(iii)
∴ ∆ABC ≅ ∆PQR …[By SSS test of similarity of triangles]
∴ ∠ABC ≅ ∠PQR = 90°
Hence Proved.