Coded Triangle Numbers
The $n$th term of the sequence of triangle numbers is given by, $t_n = \frac12n(n+1)$; so the first ten triangle numbers are: $$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$$
By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For example, the word value for SKY is $19 + 11 + 25 = 55 = t_{10}$. If the word value is a triangle number then we shall call the word a triangle word.
Using words.txt (right click and 'Save Link/Target As...'), a 16K text file containing nearly two-thousand common English words, how many are triangle words?