. Write converse of Pythagoras theorem and prove it

Best Answer

Explanation: Converse of the Pythagoras Theorem: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Given in triangle ABC, we have

AC² = AB² + BC²

Let’s construct a triangle PQR right angle at Q such that PQ = AB and QR = BC

Now, using Pythagoras Theorem in triangle PQR as angle Q = 90⁰

we have PR² = PQ² + QR²

Implies, PR² = AB² + BC² . . . (i)

Since AC² = AB² + BC² . . . (ii)

From the equatiom (i) and (ii), we have 

PR² = AB² + BC²

⇒ PR² = AC²

⇒ PR = AC

⇒ AC = PR

Now, in triangle ABC and triangle PQR, we have 

⇒ AB = PQ 

⇒ BC = QR

⇒ AC = PR

So, by SSS congruence criterion triangle ABC defnately equal to triangle PQR

Therefore, by CPCT (corresponding parts of congruent triangles are equal) 

angle B = angle Q

But, angle Q = 90⁰

Thus, angle B = 90⁰

Hence Proved. 

Final Answer: Converse of the Pythagoras Theorem: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.