Answer
$5 \text{ }cm\text{ and } 5\sqrt{3}\text{ } cm$
Work Step by Step
Let $a$ be the leg opposite the $30^o$ angle, $b$ be the leg opposite the $60^o$ angle, and $c=10$ be the hypotenuse. Using the properties of a $30-60-90$ degree triangle, then $a=5$ (half the hypotenuse) and $b=5\sqrt{3}$ ($\sqrt{3}/2$ times the hypotenuse.)
Hence, the legs measure $
5 \text{ }cm\text{ and } 5\sqrt{3}\text{ } cm
.$
