Since you have one angle and you know that the triangle is isosceles (meaning the other two angles will be equal to each other), we can figure out that the other two angles are both 49 (180 - 82 = 98. 98 divided by 2 = 49).
This is what you should be looking at now:
(Kind of messy, but you get the idea, haha.)
The line down the middle is the Height (represented by h). The distance from the bottom to the top - that's what you have to find.
Also, note that the base is 15. That means that both halves have a base of 7.5.
Since we have all our angles and measures now, we can easily find the h by working one side of the split triangle. Turn it into a triangle of its own:
It has a right angle, 90, and its B angle of 49. This means that the A angle is now 41.
Now we can use the formula a^2 + b^2 = c^2 to get h.
10^2 + 7.5^2 = h^2.
156.25 = h^2.
h = 12.5. And that's your answer.