In a right triangle, can $a+b=c?$

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.

(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)

From the pythagorean theorem, $a^2+b^2=c^2$.

Suppose $a+b=c$, then

$$ a^2+b^2=(c)^2\iff a^2+b^2=(a+b)^2 $$

Can you take it from here?