Geometry- answer remaining side and two angles

A^2+B^2=C^2
That's Pythagorean theorem.

It gave you C and A (C is hypotenuse) and A and B don't matter so I chose to use A.

X= line BC (or just B in the equation)
5^2+X^2=10^2
10^2-5^2=X^2
100-25=X^2
75=X^2
Sqrt(75)=X
X=8.660254037844386
X is approximately 8.66 (or 9 when rounded to the nearest number)

Angles are 30 degrees and 60 degrees. This is because you can use one side (let's say angle C), which is unknown, and SOHCAHTOA.
SOH in that is sine(picked angle)=Opposite/Hypotenuse

So the opposite side of angle C is 5 and the hypotenuse is 10.
Next, we plug sin^-1(5/10) to get 30.
(sin^-1 is inverse of sin)

Knowing one side is 90 and the other is 30 makes 120 degrees. Triangles have 180 degrees so 60 is left over and 60 is the measurement of angle C.

Missing side is about 8.66 or rounded to 9, full decimal places shown above if needed, Angle C is 60 degrees and Angle A is 30.

Confused? Go through the explanations again or comment.