The angles of a convex pentagon are in AP. Then the minimum possible value of the smallest angle is?

Question

The angles of a convex pentagon are in AP. Then the minimum possible value of the smallest angle is?

I tried drawing the figure and was getting 30°, but the correct answer is 36°.

Answer 1

Say the angles are $a,a+d,a+2d,a+3d,a+4d$ degrees, then we have $5a+10d=540$ or $a+2d=108$. We can thus rewrite the angles as $108-2d,108-d,108,108+d,108+2d$, and since the pentagon is convex we must have $108+2d\le180$ or $d\le36$. The largest value of $d$, 36°, will lead to the minimum smallest angle, which is also 36°.

Answer 2

The five angles are $(108-2d)^\circ$, $(108-d)^\circ$, $108^\circ$, $(108+d)^\circ$ and $(108+2d)^\circ$.

As the pentagon is convex, $108+2d<180$ and hence $2d<72$. So the smallest angle is

$$(108-2d)^\circ>108^\circ-72^\circ=36^\circ$$

$36^\circ$ is the limiting case.