Answer
$5\sqrt{2} \text{ and } 5$
Work Step by Step
Since the acute angle of the right triangle is $45^o,$ then the right triangle is isosceles. Hence, the legs both measure $5$ units.
Let the right triangle have $a$ and $b$ as the legs and $c$ as the hypotenuse. Using $a^2+b^2=c^2$ or the Pythagorean Theorem, with $
a=5
$ and $
b=5
,$ then
\begin{array}{l}\require{cancel}
a^2+b^2=c^2
\\\\
5^2+5^2=c^2
\\\\
25+25=c^2
\\\\
50=c^2
\\\\
c=\sqrt{50}
\\\\
c=\sqrt{25\cdot2}
\\\\
c=\sqrt{(5)^2\cdot2}
\\\\
c=5\sqrt{2}
.\end{array}
Hence, the missing sides have measure $
5\sqrt{2} \text{ and } 5
$ units.
