Question

prove that a triangle must have at least one acute angle.....

Answer 1

the sum of the three angles of a triangle is always 180
Solution:-
Sum of the three angles of a triangle = 180°
Let us assume that only one angle have to be acute i.e. less than 90°
Then the other two angles will have to be right or obtuse angles (90° or more than 90°)
Let us assume that both the angles are right angles (which is not possible)
Let the third angle be 'x', which is an acute angle.
⇒ 180° = x + 90° + 90°
⇒ x = 180° - 180°
x = 0
We know that an angle of a triangle cannot be 0.
Hence our assumption is completely wrong that two angles of a triangle are right angles. It cannot be possible.
Now, we can say that a triangle must have at least two acute angles.
Hence proved.

Answer 2

if a triangle does not have any atleast one acute angle it will not satisfy the property of the angle sum .
if we take the least obtuse angle 91°
and apply the angle sum property of a triangle

91°+91°=182°
which is greater than 180 even without the acute angle
thus a triangle must have atleast 2 acute angles