Q. 2

# If one angle of a triangle is equal to the sum of the other two angles, then the triangle is

A. an isosceles triangle

B. an obtuse triangle

C. an equilateral triangle

D. a right triangle

Answer :

Let us draw a triangle ABC.

We know, sum of all the angles of a triangle is 180°.

So, ∠A + ∠B + ∠C = 180° - - - - (i)

It is also mentioned in the question that one angle of the triangle is equal to the sum of the other two triangle.

Let us assume that, ∠A + ∠B = ∠C - - - - (ii)

From equation (i), we can write,

∠A + ∠B + ∠C = 180°

From equation (ii), the above equation can be written as

∠A + ∠B + ∠A + ∠B = 180°

⇒ 2 × (∠A + ∠B) = 180°

⇒ ∠A + ∠B = 90° - - - - (iii)

From equation (ii), we have

∠A + ∠B = ∠C

So, from equation (iii), we get

∠A + ∠B = ∠C = 90°

From above, it is clear that one angle of the triangle, here ∠C, is a right angle, since its value is 90°, and this is equal to the sum of the other two angles.

Therefore, this type of triangle whose one angle is equal to the sum of the other two angles is a right triangle.

Thus, the option (D) is correct.

Option (A) is not correct because in an isosceles triangle at least two sides of the triangle are equal. This implies that the two opposite angles should also be equal. It is given that one angle of the triangle, i.e., ∠C is equal to the sum of the other two angles, i.e., ∠A and ∠B. Only one case exists where ∠A = ∠B = 45°, such that ∠A + ∠B = ∠C = 90°. In all other values for ∠A, ∠B, and ∠C, the equation doesn’t hold true. So, it is not an isosceles triangle.

Option (B) is not correct because in an obtuse triangle, one of the angles is an obtuse angle, i.e., greater than 90°. But, we find out from the result that one of the angle, i.e., ∠C is 90° and the other two angles, i.e., ∠A and ∠B are the sum of the third angle, ∠C which is a right angle. Since, the sum of the two angles is equal to the third angle and the third angle is 90°, therefore neither the first nor the second angle can be greater than 90°. As none of the three angles of the triangle is more than 90°, it is not an obtuse triangle.

Option (C) is not correct because in an equilateral triangle, all the three sides of the triangle are equal. This implies, in an equilateral triangle, all the three angles are equal. But it is given to us that one angle of the triangle is equal to the sum of the other two angles where we have found one angle, i.e., ∠C as right angle. We know that sum of the angles of a triangle is equal to 180°. Since, one of the angle of the triangle, ∠C is 90°, this means that the none of the other two triangles is equal to 90°. This is in accordance with the linear pair axiom. As the three angles of the triangle are unequal, it is not an equilateral triangle.

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