The angles of a triangle are such that one angle is the average of the other two. What are the angles of the triangle?

1) π/6, π/3, π/2
2) π/3, π/3, π/2
3) π/6, π/3, π/4
4) π/2, π/2, π/3

Question

The angles of a triangle are such that one angle is the average of the other two. What are the angles of the triangle?

1) π/6, π/3, π/2
2) π/3, π/3, π/2
3) π/6, π/3, π/4
4) π/2, π/2, π/3

Accepted Answer

Let the three angles be A, B, and C such that one angle (say, B) is the average of the other two:  
B = (A + C) / 2

Also, sum of angles in a triangle:
A + B + C = π

Substitute B:
A + (A + C)/2 + C = π  
Multiply both sides by 2:  
2A + (A + C) + 2C = 2π  
2A + A + C + 2C = 2π  
3A + 3C = 2π  
A + C = 2π/3

Now, B = (A + C)/2 = (2π/3)/2 = π/3

So the angles are:  
A, B = π/3, C, with A + C = 2π/3

Let’s check the options:

Option 1: π/6, π/3, π/2  
Sum = π/6 + π/3 + π/2 = (1+2+3)/6 π = 6/6 π = π  
Check if one is average of other two:  
π/3 = (π/6 + π/2)/2 = (4π/6)/2 = (2π/6) = π/3

### Final Answer:
**1) π/6, π/3, π/2**

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