Count the number of isosceles triangles How can I count the amount of isosceles triangles fitting by the following criteria: the base of the triangles equals 1, all the angles are natural numbers and are less than $75^\circ$. (The equilateral triangle is counted only once)
If each of the base angles is $n^\circ$ and the third angle is $(180-2n)^\circ$ then you want 


$n$ an integer
$n \lt 75$ 
$180 -2n \lt 75$, i.e. $n \gt 52.5$
If the angle opposite the base is $x$ degrees, the angles at the base are $\frac{180-x}{2}$. Do a little arithmetic to determine when that is less than $75$, that will give you bounds on $x$ then it's simple to figure out how many integer values $x$ can take.