Question

Prove that a triangle must have atleast two acute angles.

Answer 1

Given ΔABC is a triangle.

To prove ΔABC must have two acute angles

Proof Let us consider the following cases

Case I: When two angles are 90°.

Suppose two angles are ∠B = 90° and ∠C = 90°

We know that, the sum of all three angles is 180°.

∴  ∠A + ∠B + ∠C = 180°   ...(i)

∴ ∠A + 90° + 90° = 180°

⇒ ∠A = 180° – 180° = 0

So, no triangle is possible.

Case II: When two angle are obtuse.

Suppose two angles ∠B and ∠C are more than 90°.

From equation (i)

∠A = 180° – (∠B + ∠C) = 180° – (Angle greater than 180°)   ...[∵ ∠B + ∠C = more than 90° + more than 90° = more than 180°]

∠A = negative angle, which is not possible.

So, no triangle is possible.

Case III: When one angle in 90° and other is obtuse.

Suppose angle ∠B = 90° and ∠C is obtuse.

From equation (i),

∠A + ∠B + ∠C = 180°

⇒ ∠A = 180° – (90° + ∠C)

= 90° – ∠C 

= Negative angle  ...[∵ ∠C in obtuse]

Hence, no triangle is possible.

Case IV: When two angles are acute, then sum of two angles is less than 180°, so that the third angle is also acute.