Question

prove that a traingle must have atleast 2 acute angles

Answer 1

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

A triangle must have two acute angle as-

1) If the triangle is a right angled triangle then 1 of the angle will be 90 degree and the sum of other two angles will be 90 degree.

So, both the other two angles have measure less than 90 degree.

2) Suppose a triangle is an obtuse triangle then take its obtuse angle as 135 degree. Then the measure of the two angles combined is 45 degree which smaller than 90 degree.

Thus proved that atleast two of the angle of the triangle are acute.

Measure of all angles of a triangle is 180 degree.