First consider a few facts about an isosceles triangle:

1) An isosceles triangle has two equal angles and one other angle. As in any triangle, the sum of these angles equals 180*.

2) As in any triangle, the largest angle will be
opposite the hypotenuse (the longest side of the triangle). This makes sense if you just think about how "open" the angle will need to be to allow for the longest side of the triangle. Thus, in an isosceles triangle, the two equal angles will be at the ends of the hypotenuse, and the other angle will be larger and opposite the hypotenuse.

Now approach the problem. We are given that the ratio of two angles in this isosceles triangle is 1:3. We must have another angle that equals either the "1" or the "3" in that ratio. Having two "3"s would not satisfy the above stated facts of an isosceles triangle (two angles must be equal and smaller than the third angle). Thus, our triangle has angles in the ratio of 1:1:3.

That totals 5 parts (1+1+3). We divide 180* by 5 (= 36*) and multiply that by 3 (the proportion of the total represented by the largest angle) in order to find the answer: 108*.