$I.$ $a, b, c, d, e$ are the measures of angles of a convex pentagon in degrees
$II$. $a \leq b \leq c \leq d \leq e$
$III.$ $a, b, c, d, e$ are in arithmetic progression is
We have,
$I$. $a, b,, c, d, e$ are angle of convex pentagon in degree.
$II$. $a \leq b \leq c \leq d \leq e$
$IIl$. $a, b, c, d, e$ are in $AP$
$a + b + c + d + e=540^{\circ}$
Let $a=\alpha$, common difference $=D$
$\therefore \quad \frac{5}{2}(2 a+4 D)=540^{\circ}$
$a+2 D=108$ and $a+4 D<180^{\circ}$
$[\because$ interior angle of polygon is
less than $\left.180^{\circ}\right]$
$\therefore \quad 108^{\circ}-2 D+4 D < 180^{\circ}$
$2 D < 180^{\circ}-108^{\circ}$
$0 < D < 36$
$\therefore$ Total 36 types are possible.
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