c^2 = a^2 + b^2 - 2 * a * b * Cos(C)

Where a, b, and c are the lengths of the sides of the triangle and A, B, and C are the angles opposite the corresponding sides. For example, A is the angle opposite side a.

Solving for C gives:

    C = Arccosine((c^2 - a^2 - b^2) / -2 * a * b)

Microsoft's Derived Math Formula Web page gives this formula for Arccosine:

    Arccosine(X) = Atn(-X / Sqr(-X * X + 1)) + 2 * Atn(1)

Putting all this together lets us find the angle between two line segments. The GetAngle function calculates the triangle side lengths. It uses the formula above and the Acos function to calculate the angle.

Public Function GetAngle(ByVal Ax As Single, ByVal Ay As _
    Single, ByVal Bx As Single, ByVal By As Single, ByVal _
    Cx As Single, ByVal Cy As Single) As Single
Dim side_a As Single
Dim side_b As Single
Dim side_c As Single

    ' Get the lengths of the triangle's sides.
    side_a = Sqr((Bx - Cx) ^ 2 + (By - Cy) ^ 2)
    side_b = Sqr((Ax - Cx) ^ 2 + (Ay - Cy) ^ 2)
    side_c = Sqr((Ax - Bx) ^ 2 + (Ay - By) ^ 2)

    ' Calculate angle B between sides ab and bc.
    GetAngle = Acos((side_b ^ 2 - side_a ^ 2 - side_c ^ 2) _
        / (-2 * side_a * side_c))
End Function

' Return the arccosine of X.
Function Acos(ByVal X As Single) As Single
    Acos = Atn(-X / Sqr(-X * X + 1)) + 2 * Atn(1)
End Function

Note also that this function always returns a positive angle. If you need to know the angle's sign, you'll need to do a little more work.