Question

prove that a triangle must have atleast two acute angles

Answer 1

Prove that a triangle must have at least two acute angles.
Answer 6:
Given:
∆ is a triangle
To Prove:
∆ must have two acute angles.
Proof:
Let us consider the following cases
Case I
When two angles are 90°
Suppose two angles are ∠ = 90° and ∠ = 90°
We know that, the sum of all three angles is 180°
∴ ∠ + ∠ + ∠ = 180°
∴ ∠ + 90° + 90° = 180°
⇒ ∠ = 180° − 180° = 0, which is not possible.
Hence, this case is rejected.
Case II
When two angle are obtuse.
Suppose two angles ∠ ∠ are more than 90°
We know that, the sum of all three angles is 180°
∴ ∠ + ∠ + ∠ = 180°
∠ = 180° − (∠ + ∠) = 180° − (Angle greater than 180°)
∠ = negative angle, which is not possible.

Answer 2

Given:
∆ is a triangle
To Prove:
∆ must have two acute angles.
Proof:
Let us consider the following cases
Case I
When two angles are 90°
Suppose two angles are ∠ = 90° and ∠ = 90°
We know that, the sum of all three angles is 180°
∴ ∠ + ∠ + ∠ = 180°
∴ ∠ + 90° + 90° = 180°
⇒ ∠ = 180° − 180° = 0, which is not possible.
Hence, this case is rejected.
Case II
When two angle are obtuse.
Suppose two angles ∠ ∠ are more than 90°
We know that, the sum of all three angles is 180°
∴ ∠ + ∠ + ∠ = 180°
∠ = 180° − (∠ + ∠) = 180° − (Angle greater than 180°)
∠ = negative angle, which is not possible.
Hence, this case is also rejected.
Case III
When one angle in 90° and other is obtuse.
Suppose angles ∠ = 90° and ∠ is obtuse.
We know that, the sum of all three angles is 180°
∴ ∠ + ∠ + ∠ = 180°
⇒ ∠ = 180° − (90° + ∠)
= 90° − ∠
= Negative angle, which is not possible.
Hence, this case is also rejected.
Case IV
When two angles are acute, then sum of two angles is less than180°, so that the
third angle is also acute.
Hence, a triangle must have at least two acute angles.



Hope you have helped. Thank you