If two triangles have one angle equal to one angle, the sides about other angles proportional, and the remaining angles either both less or both not less than a right angle, the triangles will be equiangular and will have those angles equal about which the sides are proportional.
Proof
Construct, at the vertex of the angle whose sides are proportional,
an angle equal to the corresponding angle in the second triangle
(I.23). The resulting auxiliary triangle agrees with the first in
two angles (and hence all three, by I.32) and shares a side with the
second; the constraint on the remaining angle being acute or obtuse
ensures the construction is non-ambiguous (essentially eliminates
the SSA failure case).