3. As shown in Figure 6, in quadrilateral $ABCD$, $BC \parallel AD$, $BC=26, AD=5, AB=10, CD=17$. If the angle bisectors of $\angle A$ and $\angle B$ intersect at point $M$, and the angle bisectors of $\angle C$ and $\angle D$ intersect at point $N$, what is the length of $MN$?
3. 2 .

As shown in Figure 10, draw perpendiculars from points $M$ and $N$ to the sides of the trapezoid, with the feet of the perpendiculars being $O, P, Q, R, S, T$.
By the angle bisector property, we have
$$
M O=M P=M Q, N R=N S=N T .
$$

Since $A D \parallel B C$, then
$$
M P=M R, M O=M S
$$
$\Rightarrow$ quadrilaterals $M N R P$ and $M O S N$ are both rectangles. Combining with the angle bisector property, we get
$$
A P=A Q, B O=B Q, C S=C T, R D=D T \text {. }
$$

Therefore, $M N=\frac{1}{2}(B C+A D-A B-C D)=2$.