Brain-Flak, 42 bytes

Here's a 46 byte solution I like more.

Brain-Flak, 40 bytes

(([{}](((()))<>))<>){<>({}({}({})))}{}{}

Wheat Wizard and I had a duel over this question. When we decided to post our solutions we were tied at 42 bytes, but I found a 2 byte golf of his solution. We decided that would count as the tie breaker (my solution is below).

Try it online!

Explanation:

# Set up the stacks like this:  -input
                                     1     -input
                                     1          1
(([{}](((()))<>))<>)                 ^

# Output 1 for triangular and 0 for non-triangular 
{<>({}({}({})))}{}{}

For a full explanation please see Wheat Wizard's answer.

Brain-Flak, 42 bytes

Here's a 46 byte solution that I found interesting.

Try it online! (truthy)
Try it online! (falsy)

The idea is to subtract 1 then 2 then 3 all the way up to the input. If you hit 0, then you know this is a triangular number, so you can stop there.

Try it online! (truthy)
Try it online! (falsy)

# Push -input on both stacks. One is a counter and the other is a running total
(([({})])<>)

# Count up from -input to 0
{
  # Push the new total which is: (counter += 1) + total (popped) + input (not popped)
  # This effectively adds 1, then 2, then 3 and so on to the running total
  (({}())<>{}({}))
  # If not 0
  {
    # Push to 0s and switch stacks to "protect" the other values
    ((<>))
  # End if
  }
  # Pop the two 0s, or empty the stack if we hit 0
  {}{}
# End loop
}

Here's a 46 byte solution I like more.

{<>(({}())){({}[()]<>{(<({}[()])>)}{}<>)}{}<>}

Outputs 0\n (literal newline) for truthy, the empty string for falsy.

The idea is to count down from input by consecutive numbers, 1 at a time. E.g. input - (1) - (1,1) - (1,1,1). Each time we subtract, if we aren't at 0 yet, we leave an extra value on the stack. That way, if we are at 0 and are still subtracting when we pop we remove the last value on the stack. If the input was a triangular number, we will end exactly at 0, and wont pop the 0.

Try it online! truthy
Try it online! falsy

# Implicit input (call it I)

# Until we reach 0, or the stack is empty
{
  # Add 1 to the other stack and push it twice. This is our counter.
  <>(({}()))
  # While counter != 0
  {
    # counter -= 1
    ({}[()]
    # if I != 0 
    <>{
      # I -= 1, and push 0 to escape the if
      (<({}[()])>)
    # End if
    }
    # Pop from the stack with I. This is either the 0 from the if, or I
    {}
    # Get ready for next loop End while
    <>)
  # End While
  }
  # Pop the counter that we were subtracting from
  {}<>
# End Until we reach 0, or the stack is empty.
}