Help explaining minor keys: diminished and half-diminished chords?

Question:

Help explaining minor keys: diminished and half-diminished chords?

In my music theory class we just finished up talking about major chords and have moved on to minor keys and the chords involved in them.

According to my understanding the following chords are in a minor key:
i, ii (diminished), III, iv, v7, VI, VII7 (half-diminished)

Why is the VII half-diminished? How does one go about writing a half-diminished chord? I understand that a diminished chord has two minor 3rds stacked on top of each other. Keeping this in mind, does this mean that a regular VII triad would be diminished?

And since you can have a half-diminished chord can you have a half-augmented chord? Could a triad ever be half-diminished?

Any imput would be greatly appreciated! Thanks!

Best Answer - Chosen by Asker: Let's say we're in the key of a minor, with no sharps or flats...
the vii7 would be made up of G#,B,D,F. You would use G# as a raised leading tone, so you would have to be in harmonic minor, not natural or melodic.

G#-B is a minor triad
B-D is a minor triad
D-F is a minor triad
G#- D is a diminished fifth

It is the minor third, diminished fifth, and most importantly- the MINOR seventh, that makes the vii7 in harmonic minor a 'half diminished chord'. In order for it to be fully diminished, you would need a diminished seventh of the chord, making it G#,B,D,Fb.

A regular vii triad would be diminished- G#,B,D

A triad can never be half-diminished because it is the seventh of the chord (fourth note) that makes it half-diminished.

An augmented triad would be (for example, in the key of C major, no sharps or flats) C,E,G#. If you continue the pattern of stacking major thirds, making it C,E,G#,B#, you simply have an octave on top. (B#=Cnatural) In terms of half-augmented...no, because if you had the augmented traid (C,E,G#), and a major seventh on top, it would simply be C,E,G#,B, or basically a CmajorSeventh Chord with a sharp fifth. At that point, you're really heading out of tonal harmony and hitting some pretty jazzy stuff.