Consider this diagram. Half of the line segments are drawn in red. The line segments $a_0b_1,b_1a_2,a_2b_3,b_3a_4$ also exist, but are ommited here for clarity.
We do not yet know what angle $x$ is, but we can work out the other angles in terms of $x$. We will be working in radians here. $\pi\ radians = 180\ degrees$
Because the slice is an isosceles triangle, we can calculate:
$c = \frac{\pi-x}{2} = 0.5\pi - 0.5x$
Knowing that all the inner triangles are also isosceles trangles, we can also calculate:
$d = \pi - 2c = x$
$e = c - d = 0.5\pi - 1.5x$
$f = \pi - 2e = 3x$
$g = \pi - c - f = 0.5\pi -2.5x$
$h = \pi - 2g = 5x$
$i = \pi - e - h = 0.5\pi - 3.5x$
$j = \pi - 2i = 7x$
Also because of the isosceles triangle, $x = i$, so we can now derive the value of $x$:
$x = 0.5\pi - 3.5x$
$4.5x = 0.5\pi$
$9x = \pi$
Now we can go back and substitute $\frac{\pi}{9}$ for $x$ and work out all the angles exactly.
We can also see a pattern in the angles. The angles of the tops of the triangles go:
$x,3x,5x,7x..$
while the angles of the sides go
$0.5\pi-0.5x$
$0.5\pi-1.5x$
$0.5\pi-2.5x$
$0.5\pi-3.5x$
$..$
So we can work out the angles of specific triangles with closed form formula, instead of having to calculate them all.