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  • Prove that a triangle must have at least two acute angles.

Prove that a triangle must have at least two acute angles.

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Solution.

It is given that a triangle must have at least two acute angles.
An acute angle is less than 90 degrees
Let us assume that a triangle does not have two acute angles.
So, it has two angles that are either right angles (=90 degrees) or obtuse angles (greater than 90 degrees)
So let two right angles are present,
So using angle sum property of a triangle, third angle must be zero which is not possible.
Also let one angle be right and one be obtuse. We can take the smallest obtuse angle, i.e.,
So using angle sum property of a triangle, third angle must be negative which is not possible.
 Again, if both the angles are obtuse the third angle must be negative which is not possible.
So it is necessary for a triangle to have at least two acute angles.

Hence proved

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