just go through and check the definitions.
we have x + y = 0
=> x = -y
(i) reflexive means (x,x) is in R. does x = -x? No, not in general. this only happens if x = 0, and so it does not work for all real numbers. the relation is not reflexive.
(ii) symmetric means (x,y) is in R implies that (y,x) is in R.
Assume (x,y) in R, then x = -y. Multiplying both sides by -1 we get y = -x, that is, (y,x) is in R. The relation is symmetric.
(iii) anti-symmetric means if (x,y) and (y,x) is in R, then x = y.
The relation is not anti-symmetric. Counter-example, (1,-1) is in R. Clearly x = -y and y = -x but x and y are distinct.
(iv) transitive means if (x,y) is in R and (y,z) is in R, then (x,z) is in R.
The relation is not transitive. Counter-example: (1,-1) and (-1,1) is in R, but (1,1) is not.
Now do the others the same way. let the definitions guide you