Two ways to do it. 

1. An easy way that you can do in your head,
but which your teacher won't allow, but it's good
for checking your answer:

Half of 37 is 18.5, so the two numbers are the two
closest whole numbers to 18.5. So the answer: 18 and 19.

2. By algebra:

The smaller = x
The larger = x+1

Their sum = x + (x+1) 

Their sum = 37

  x + (x+1) = 37
  x + x + 1 = 37
     2x + 1 = 37
         2x = 36
          x = 18 = the smaller

The larger = x+1 = 18+1 = 19.

Edwin