#### Answer

$\dfrac{13\sqrt{2}}{2} \text{ and } \dfrac{13\sqrt{2}}{2}$

#### Work Step by Step

Since the given acute angle of the right triangle is $45^o,$ then the the legs ($a$ and $b$), measure $\dfrac{\sqrt{2}}{2}\cdot c,$ where $c$ is the hypotenuse. Hence, the legs measure
\begin{array}{l}\require{cancel}
\dfrac{\sqrt{2}}{2}\cdot c=a
\\\\
\dfrac{\sqrt{2}}{2}\cdot 13=a
\\\\
\dfrac{13\sqrt{2}}{2}=a
\\\\
a=\dfrac{13\sqrt{2}}{2}
.\end{array}
Hence, the missing sides have measure $
\dfrac{13\sqrt{2}}{2} \text{ and } \dfrac{13\sqrt{2}}{2}
$ units.