Question

The measures of the angles of a triangle are in A.P. and the angle with largest measure has radian measure $\frac{5\pi }{12}$. then find the measure of the angle with smallest measure.

Answer 1

Let the angles of the triangle is $a-d,a,a+d$

Since we know that sum of the angle of the triangle is 180^0

Thus,

$a-d+a+a+d={180}^0$

$3a={180}^0$

$a={60}^0=\frac{\pi }{3}$

We have the highest angle measure is$\frac{5\pi }{12}$

So,

$\frac{5\pi }{12}=a+d$

$\frac{5\pi }{12}-\frac{\pi }{3}=d$

$d=\frac{\pi }{12}$

Thus the measure of the shortest angle is:

$a-d$

$\frac{\pi }{3}-\frac{\pi }{12}$

$=\frac{3\pi }{12}$