Replacing $x,y$ in $f(x,y,z)$ with values from a list

I’m sure there is a simple solution to this using map or apply, but its not occurring to me.

Suppose I have a function $ f(x,y,z)= x+y+z)$

And I want to evaluate $ f(x,y,1)$ for $ \{x,y\} ∈ \{\{1,2\},\{3,4\}\}$

What is the best way to do this?


Examples:

If I use f[x_,y_,z_]:= x+y+z; f[#1,#2,1]&/@ {{1,2},{3,4}}

this will give me things like {{2+#2,3+#2}}.

On the other hand, Apply works on a single element f[#1,#2,1]& @@ {1,2} but not on a list such as {{1,2},{3,4}}. On the list it gives me {f[1,3,1],f[2,4,1]

I am not sure how to go from the case of a single pair to a list of pairs