What is the least number of acute angles a triangle can have?
The Sum of interior angles of a polygon is always (n - 2) ×180º.
Answer: A triangle must have at least 2 acute angles in it.
The Sum of interior angles of the triangle is 180º
Sum of interior angles of a triangle = 180º [Angle sum property of a triangle]
Since there are three interior angles present inside the triangle. So the sum of three angles needs to be exactly equal to 180º.
Let the three angles of a triangle be A, B, and C, respectively.
Thus, A + B + C = 180º ----------(1)
Consider Case 1) In a right-angled triangle, with B = 90º
Thus equation (1) reduces to A + C = 90, so A and C cannot be obtuse( greater than 90º), so they must both be acute only.
Case 2) When we suppose two obtuse angles in a triangle.
This case violates our equation 1 since 2 obtuse angles will have a sum greater than 180 degrees. So such a possibility is neglected as it cannot occur.
Case 3) When we take 2 acute angles in the triangle.
Now let A, B be acute angles, thus the sum of B will be less than 180º, thus Angle C can have any value( either acute or even obtuse).
So it can be easily concluded from case 3 that in each triangle, there are at least 2 acute angles.