Here's the diagram illustrating the angle bisector in triangle $ABC$. As shown:

• The line $AD$ bisects $\angle A$, meeting side $BC$ at point $D$.
• The segments $BD = 4$ units and $DC = 6$ units are shown, with side lengths $AB = 6$ units and $AC = 9$ units.
• The angle bisector theorem is reflected in the proportions, where $\frac{BD}{DC} = \frac{AB}{AC} = \frac{6}{9} = \frac{2}{3}$.

Let me know if you'd like to dive deeper into any part of the explanation, or if you have more questions!

Here are some related questions to further explore:

1. What happens if the angle bisector does not divide the opposite side proportionally?
2. How can the angle bisector theorem help in solving real-world problems?
3. What if the triangle was isosceles—how would the angle bisector behave?
4. How do you use the angle bisector theorem in proving similarity of triangles?
5. What are other applications of proportional relationships in geometry?

Tip: When using the angle bisector theorem, always check for any given side lengths or angles that may simplify the math!