What are the angles of spherical triangles of a sphere partitioned in 4 equal spherical triangles?

Question

A tetrahedron inside a sphere can divide a sphere into 4 equal spherical triangles.

What are the angles, coordinates of vertices and arc lengths of those spherical triangles?

Bear in mind link:
Since the sides of a spherical triangle are arcs, they can be described as angles, and so we have two kinds of angles:

1. The angles at the vertices of the triangle, formed by the great circles intersecting at the vertices and denoted by Greek letters.
2. The sides of the triangle, measured by the angle formed by the lines connecting the vertices to the center of the sphere and denoted by lower-case Roman letters.

Answer 1

Answering your title question, the angles are, by symmetry, $2\pi/3$.