If the square of a side of any triangle is equal to the sum of the squares of other two sides, the angle between the latter two sides is a right angle.
Proposition:
Let in ΔABC, AB2 = AC2 + BC2
It is required to prove that ∠C is a right angle.
Construction:Draw a triangle ΔDEF so that ∠F = 1 right angle.
EF = BC and DF = AC.
Proof:DE2 = EF2 + DF2 [Since in ΔDEF, ∠F is aright angle]
= BC2 + AC2 = AB2
∴ DE = AB
Now, in ΔABC and ΔDEF , BC = EF, AC = DF and AB = DE. [supposition]
∴ ΔABC ≅ ΔDEF; ∴ ∠C = ∠F
∴ ∠F =1 right angle.
∴ ∠C= 1 right angle. (Proved)
Proposition:
Let in ΔABC, AB2 = AC2 + BC2
It is required to prove that ∠C is a right angle.
Construction:Draw a triangle ΔDEF so that ∠F = 1 right angle.
EF = BC and DF = AC.
Proof:DE2 = EF2 + DF2 [Since in ΔDEF, ∠F is aright angle]
= BC2 + AC2 = AB2
∴ DE = AB
Now, in ΔABC and ΔDEF , BC = EF, AC = DF and AB = DE. [supposition]
∴ ΔABC ≅ ΔDEF; ∴ ∠C = ∠F
∴ ∠F =1 right angle.
∴ ∠C= 1 right angle. (Proved)
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