let the interior angle of each polygon be $x$. notice the two adjacent angles to the 72 degree angle, all of them sum to 180. so we have $2x + 72 = 360.$ this yields $x = 144.$

let the number of sides of the polygon be $n$. we have the total sum of interior angles of the polygon equal to both $180(n-2)$ and $144n$

setting these equal to each other, we have $180(n - 2) = 144n.$ this yields $n = 10.$ so, your answer is $\boxed{10}$ sides.

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