Yes but you aren't obligated to call the turn if a truck hits.
If you do plan on seeing the river card, you can flop-check, turn call 1 BB (total cost $10); flop-bet, turn check/check (total cost $5) or flop-bet and call reraise, call turn bet (total $20) . Or even, and I'm not saying this is likely, Check-flop, check/check- turn (total cost $0)
So best case you can see the river for $0, worst case for $20. The only one that costs you $20 is betting the flop.
And please don't think I'm saying you are wrong. I'm probably totally off here, but since I only have ace high here I want to see the next card(s) as cheaply as possible.