4. Through the vertex of isosceles $\triangle A B C$, draw a line intersecting the extension of the opposite side at $D$. If the resulting triangles are all isosceles, then $\triangle A B C$ has $\qquad$ such configurations.
4.5.

Draw a straight line from the base angle to intersect with the extension of the opposite side, there are 5 $\triangle A B C$ that meet the conditions, with base angles of $45^{\circ} 、 72^{\circ} 、 36^{\circ} 、 \frac{180^{\circ}}{7}$ 、 $\frac{2 \times 180^{\circ}}{7}$.
Draw a straight line from the vertex angle to intersect with the extension of the opposite side, there is 1 $\triangle A B C^{\prime}$ that meets the conditions, with a base angle of $72^{\circ}$, which is repeated above.