Short Answer: it depends on the surface (the geometry) you're working on.
Long Answer:
Historically, $(2D)$ geometry and its laws have been made with the assumption that the objects lie on a plane. This means that the surface all objects under consideration are embedded in is flat, and we get nice, neat Euclidean geometry. This is the geometry most people are familiar with, as it closely resembles the geometry of our world on a small scale.
However, this geometry relies on 5 axioms which are mostly self-evident. The fifth one, called the parallel postulate, is equivalent to the property that the sum of angles in a triangle is 180 degrees. Whenever we're working in this geometry, this must be true, and it is satisfied in a way in a triangle with two perpendicular angles. To see why, look at this diagram:
In an isosceles triangle, the height from the apex is also the angle bisector, hence the angle $a$ is divided into two equal parts. We want to see the behavior of $a$ as the equal sides get longer and longer. In each right triangle formed, we have the equal side as the hypotenuse, so we'll call it $h$. Similarly, half the base is the side opposite the angle $a/2$, and it'll be denoted by $o$. The relation between the sides and the angle is (by definition of sine):
$$\sin{(a/2)} = \frac{o}{h}$$
Using the double angle formula for cosine to find an expression in terms of $a$, we get:
$$\sqrt{\frac{1 - \cos{(a)}}{2}} = \frac{o}{h}$$
I dropped the $\pm$ since the sides are always positive. Now, we solve for $a$ to get a formula for the angle from the two sides:
$$a = \cos^{-1}{\left(1 - \frac{2o^2}{h^2}\right)}$$
From here, we can take the limit (if your calculus is a bit rusty, it's a tool we use to describe the behavior of functions while the variable is around a point) of the function as $h$ goes to infinity:
$$\lim_{h \to \infty} \cos^{-1}{\left(1 - \frac{2o^2}{h^2}\right)} = \cos^{-1}{(1)} = 0$$
Thus, when the hypotenuse gets infinitely long, the angle tends to 0, and the base angles are 90 degrees since the sum must be 180 degrees.
But what if the sum didn't have to be 180? In fact, there are geometries called non-Euclidean geometries where this fifth axiom is broken and the objects are on another 3D object besides a plane. In the case of spherical geometry (which is what we use over large distances on Earth since our planet is roughly a sphere), there are triangles with two right angles and a non-zero third angle, for example. Look at this drawing:
(source: scientificamerican.com)
It has three right angles, and finite side lengths. So, to sum up, in Euclidean geometry, extending the apex to infinity results in a zero angle and two 90 angles, but this is a rather degenerate case. In non-Euclidean geometries such as spherical geometry, such triangles with two 90 degree bases are common, and you can have any non-zero angle as the apex angle's measure.