In a right triangle, can $a+b=c?$
Solution 1:
Your question has nothing to do with right triangles. If $c=a+b$ in any triangle, then the vertex $C$ must lie on the segment $AB$, and this is called a "degenerate" triangle.
Solution 2:
(Addendum: As noted as well by others, this should highlight that your question has little to do with right triangles, as this is in fact a property of triangles in general)