As far as DL goes, I translate the as das/die/der (e.g. Die Tasche ist schön; Der Hund ist schön; Das Mädchen ist schön), and dies/diese/dieser/dieses to this. It's pretty safe. ☺ As far as that goes, das generally works, at least until you're at higher levels.
If you have a sentence, such as Die Tasche ist schön and a translation of The bag is nice is marked wrong (instead of This bag is nice), report it.
Well, both examples are really the same. The "meine" in "Diese Tasche ist meine" is equivalent to the "meine" in "Meine Frau ist schön". They are both feminine nouns in singular. Or so I'm thinking...
Though, I'm struggling a little now if I take a masculine noun:
"Mein Mann ist schön." - But: "Dieser Mann ist meiner."
Or, a neuter noun:
"Mein Schwein ist schön." - And: "Dieses Schwein ist mein(e)s."
Just like the other day, I was wondering:
Ein schöner Mann.
Der schöne Mann.
Gaa. What? I don't know what's going on here.
Sometimes (your own) language creates riddles for you that you don't know how to answer.
Sorry, I couldn't be of real help.
About the Ein schöner Mann vs Der Schöne Mann thing ,it used to dazzle me as well but i found a wonderful website that a member here posted ,here it is: http://yourdailygerman.wordpress.com/2012/10/08/adjective-declension-german/
Basically what that is about is that Ein does not convey full information so Schöner completes it ,but with Der it doesnt have to. Read the three part article and take the test at the end ,you would be surprised how simple they make it!