Let's walk through completing the flowchart proof based on the given information and the image.

Given:

• $BD$ bisects $\angle ABC$. This is the given information.

Flowchart Breakdown:

1. $\angle ABD \cong \angle CBD$ (Reason: Definition of Angle Bisector)

   • Since $BD$ bisects $\angle ABC$, it divides $\angle ABC$ into two equal angles: $\angle ABD$ and $\angle CBD$.

2. $AB \cong BC$ (Reason: Given or Reflexive Property)

   • From the image, we see that the two sides $AB$ and $BC$ are marked as congruent. This can be stated as given, or we use the fact that they are congruent from the markings.

3. Conclusion:

   • Now you can use Angle-Side-Angle (ASA) congruence to conclude that $\triangle ABD \cong \triangle CBD$.

Full Flowchart Reasoning:

• $BD$ bisects $\angle ABC$ (Reason: Given)
• $\angle ABD \cong \angle CBD$ (Reason: Definition of Angle Bisector)
• $AB \cong BC$ (Reason: Given)
• $\triangle ABD \cong \triangle CBD$ (Reason: Angle-Side-Angle (ASA) Postulate)

Would you like more details on any part of the proof?

Relative Questions:

1. How does the definition of an angle bisector help in proving triangle congruence?
2. What other congruence postulates can be used for proving triangles congruent besides ASA?
3. How does marking congruent sides visually in a diagram help in solving proofs?
4. What is the significance of the Angle-Side-Angle postulate in geometry?
5. Can you use the Side-Angle-Side postulate in this proof instead of ASA?

Tip: Always mark the diagram with given information, like congruent sides or bisected angles, to make proofs easier to visualize.