Your description doesn't make sense. Please provide a picture.
"Opposite ray" has no meaning even if "ray AC" means the side AC (extended) of the triangle ABC.
Half plane suggests a three-dimensional structure, whereas other parts of the question suggest a plane figure. Which is it? 3-D or 2-D?
x appears to be a point which is a vertex of another triangle CDx. Does "CDx equals ABC" imply two congruent triangles, or triangles which have equal area? Or is it the measure of two angles in these triangles? If congruent, what are the corresponding sides, since there's only one common vertex (C)?
If triangles ABC and CDx are in different planes, are these planes orthogonal? Are these triangles in the same plane?