# . Write converse of Pythagoras theorem and prove it

Write converse of Pythagoras theorem and prove it.

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.