Step 1: Understanding via Direction Ratios:
The direction ratios of a line through two points are the differences of their corresponding coordinates.
Two lines are collinear when the direction ratios of one line are proportional to the direction ratios of the other.
Step 2: Find the Direction Ratios:
Points are $A(1,1,1)$, $B(2,5,0)$, $C(3,2,-3)$, $D(1,-6,-1)$.
Direction ratios of line AB are obtained by subtracting coordinates of A from B, and similarly for CD.
\[ AB: (2-1,\,5-1,\,0-1)=(1,4,-1), \qquad CD: (1-3,\,-6-2,\,-1-(-3))=(-2,-8,2) \]
Step 3: Test Proportionality:
Divide each direction ratio of CD by the matching direction ratio of AB.
\[ \frac{-2}{1}=\frac{-8}{4}=\frac{2}{-1}=-2 \]
All three ratios are equal to the same constant $-2$, so the direction ratios of CD are proportional to those of AB, which proves AB and CD are collinear.
Step 4: Find the Angle Using Direction Ratios:
Use the standard cosine formula for the angle between two lines with direction ratios $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$.
\[ \cos\theta=\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}}=\frac{(1)(-2)+(4)(-8)+(-1)(2)}{\sqrt{18}\,\sqrt{72}}=\frac{-36}{36}=-1 \]
This gives $\theta=180^\circ$, and the negative proportionality constant found in Step 3 confirms the direction reversal that produces this angle.
Final Answer:
Proportional direction ratios with constant -2 prove collinearity, and the angle between AB and CD is 180 degrees.
\[ \boxed{\theta=180^\circ, \quad AB \parallel CD} \]