The angles of a triangle are in A.P., the least being half the greatest. Find the angles

In an arithmetic progression (A.P.), the difference between any two successive members is a constant. Let's denote this constant as 'd'.

Given that the angles of the triangle are in A.P., we can denote the three angles as follows:

1. The smallest angle: a - d
2. The middle angle: a
3. The largest angle: a + d

We also know that the sum of the angles in a triangle is 180 degrees. Therefore, we can write the equation:

(a - d) + a + (a + d) = 180 3a = 180 a = 180 / 3 a = 60 degrees

The problem also states that the smallest angle is half the largest. Therefore, we can write the equation:

a - d = 0.5 * (a + d) Solving this equation for 'd' gives:

d = a / 3 d = 60 / 3 d = 20 degrees

Therefore, the three angles of the triangle are:

1. The smallest angle: a - d = 60 - 20 = 40 degrees
2. The middle angle: a = 60 degrees
3. The largest angle: a + d = 60 + 20 = 80 degrees

So, the angles of the triangle are 40 degrees, 60 degrees, and 80 degrees.