Since TMTOWTDI how about this?
(not as concise, but clearer (to me) )

open( FILE, "<data.txt");
my @lines = (<FILE>);
close FILE;
my $i = 0;
for(@lines){
  chomp;
  print $_ if (  $lines[$i-1] =~ /116">$/ ) && ( $i );
  $i++;
  }
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The second clause of the if conditional is to prevent the "wrap around" effect of a -1 array subscript.