Question

show that if sum of the two angles of a triangle is equal to the third angle then the triangle is right angled triangle​

Answer 1

Let the Triangle be ABC,

The three angles are ∠A, ∠B, ∠C

By Angle sum property which states,

The sum of three angles in a triangle equals a straight line.

∠A + ∠B + ∠C = 180° _____ ( Equation 1)

Given, Sum of two angles is equal to third angles.

So, ∠A + ∠B = ∠C ________ ( Equation 2)

Now,

⇒∠A + ∠B + ∠C = 180°

⇒ ∠C + ∠C = 180° ( From Equation 2)

⇒ 2 ∠C = 180°

⇒ ∠C = 90°

So, When the sum of two angles is equal to third angle, One of the angle is 90°

If one of the angle in a triangle is 90°, Then the triangle is Right angled triangle.

Therefore, When the sum of two angles is equal to third angle, The triangle is Right angled triangle.

Answer 2

Given:

A triangle ∆ABC in which sum of the two angles of a triangle is equal to the third angle.

To prove:

∆ABC is a right angled triangle.

Proof:

To prove the statement, we are going to use angle sum property of a triangle. Let us see what this property says:

Angle Sum property says that the sum of all the angles of a triangle is 180°.

⇒∠A + ∠B + ∠C = 180°

We know that ∠A + ∠B = ∠C _____[Given]

So,

⇒ (∠A + ∠B) + ∠C = 180°

⇒ ∠C+∠C = 180° ______[∠A + ∠B = ∠C]

⇒ 2∠C = 180°

⇒ ∠C = 180/2

⇒ ∠C = 90°

We know that from three angles, if one angle is 90°, then the triangle is a right angled triangle.

Hence, ∆ABC is a right angled triangle

Proved!!