The angles of triangle are such that one is average of other two, then the…

To find the angles of a triangle where one angle is the average of the other two, we can use algebra and the fundamental properties of triangles.

Step-by-Step Derivation
Define the variables:
Let the three angles of the triangle be A, B, and C.
The sum of the angles in any triangle is always 180 degrees (or π radians):
A + B + C = 180

Apply the given condition:
One angle is the average of the other two. Let C be that angle:
C = (A + B) / 2
This implies: 2C = A + B

Solve the system of equations:
Substitute (A + B) with 2C in the sum equation:
(A + B) + C = 180
2C + C = 180
3C = 180
C = 60 degrees (or π/3 radians)

Analyze the options:
Since one angle must be 60 degrees (π/3), we check the sets provided in the image:

π/6, π/3, π/2 (30°, 60°, 90°)

π/3, π/3, π/2 (60°, 60°, 90°) -> Sum is 210° (Incorrect)

π/6, π/3, π/4 (30°, 60°, 45°) -> Sum is 135° (Incorrect)

π/2, π/2, π/3 (90°, 90°, 60°) -> Sum is 240° (Incorrect)

Looking closely at the logic provided in the solution, there seems to be a mismatch between the provided options' sum and the requirement. However, based on the question's condition that one angle is the average, 30°, 60°, and 90° (Option 1 in the original image set) is the only set that sums to 180° and satisfies 60° = (30° + 90°) / 2.