What is the least number of acute angles a triangle can have?

Solution:

The Sum of interior angles of a polygon is always (n - 2) ×180º.

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.

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What is the least number of acute angles a triangle can have?

Summary:

A triangle must have at least 2 acute angles in it.