3. As shown in Figure 1, in the geometric body $A B C-A_{1} B_{1} C_{1}$, $A B = A_{1} B_{1}$. The necessary and sufficient condition for $A A_{1}$, $B B_{1}$, and $C C_{1}$ to be concurrent is ( ).
(A) $B C \parallel B_{1} C_{1}$ and $A C \parallel A_{1} C_{1}$
(B) $B C \neq B_{1} C_{1}$ and $A C \neq A_{1} C_{1}$
(C) $A A_{1} \neq B B_{1}$ and $\angle B A A_{1} \neq \angle \dot{B}_{1} A_{1} A$
(D) Plane $A B C$ is not parallel to plane $A_{1} B_{1} C_{1}$
3. (C).

Three planes intersect each other pairwise, resulting in three intersection lines $A A_{1}$, $B B_{1}$, and $C C_{1}$, which either intersect at one point or are parallel to each other. When $A A_{1}$, $B B_{1}$, and $C C_{1}$ intersect at one point, from $A B=A_{1} B_{1}$, we know that plane $A B C$ and plane $A_{1} B_{1} C_{1}$ cannot be parallel, so (A) does not meet the necessary condition (it is also not sufficient); (B) and (D) satisfy the necessity, but when $A B / / A_{1} B_{1}$, the condition is not sufficient. Therefore, (C) should be chosen.

Example 1: In quadrilateral $A A_{1} B_{1} B$, if $A A_{1} \neq B B_{1}$, it can be a trapezoid, which is equivalent to $A A_{1}$ and $B B_{1}$ intersecting, and also implies that $A A_{1}$, $B B_{1}$, and $C C_{1}$ intersect at one point.