Obama: "voting is the best revenge."
While I was listening to President Obama, he pronounced the in this way /ðə/.
Romney: "In his closing argument president Obama asked his supporters to vote for revenge, for revenge. I ask the American people to vote for love of country."
However, Romney pronounced the in this way /ði/.
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I need a website in which I will study these sounds on my own. Thanks in advance.
You don't really need a website. In most varieties of English, /ðə/ is used before consonant sounds and /ði:/ before vowel sounds. /ði:/ is used before vowels if the word 'the' is stressed.