Proof of the Converse of Pythagorean Theorem
Given: Δ ABC with AC2 = AB2 + BC2
Converse of pythagorean theorerm
Prove: ABC = 90°
Construction: We construct a right triangle PQR, such that PQ = AB, QR = BC and PQR = 90°
Proof:
In Δ PQR, Q = 90°.
By Pythagoras theorem, PQ2 + QR2 = PR2
From construction, AB2 + BC2 = PR2 ------ (1)
But, AB2 + BC2 = AC2 ----------- (2)
From (1) and (2),
we get: PR2 = AC2
PR = AC.
By SSS criteria for congruence of triangles,
we get Δ ABC Δ PQR.
So that B = Q = 90°.