Question

The measures of the angles of a convex pentagon are in arithmetic series. The smallest angle of the pentagon:

• A. must be $>$ ${36}^{\circ }$
• B. cannot be $<$ ${36}^{\circ }$
• C. must be $={36}^{\circ }$
• D. may be of any measure

Answer 1

The correct option is C must be $={36}^{\circ }$

Let the angles of a pentagon $=a,a+d,a+2d,a+3d,a+4d$, which are in A.P.
Sum of the angles of a pentagon $=540$
$\Rightarrow 5a+10d={540}^o$
$\Rightarrow a+2d={108}^o---------------\left(1\right)$
Let the largest angle be $a+4d$.

As the pentagon is convex, $a+4d<{180}^o--------\left(2\right)$

From (1) and (2)

$216-a<{180}^o$

$a>{36}^o$