This question is asking you to complete the flowchart proof for two triangles, $\triangle ABD$ and $\triangle CBD$, with the given that $D$ is the midpoint of $AC$. Let's walk through the reasoning behind each step:

Given:

• $D$ is the midpoint of $AC$.

Step 1: $AD \cong CD$

• Reason: Since $D$ is the midpoint of $AC$, it divides $AC$ into two equal segments, so $AD = CD$.
• Reason: Definition of midpoint

Step 2: $\angle A \cong \angle C$

• Reason: $\triangle ABC$ is isosceles (as indicated by the tick marks on $AB$ and $BC$, showing that $AB \cong BC$), so the base angles $\angle A$ and $\angle C$ are congruent.
• Reason: Base angles of an isosceles triangle are congruent

Step 3: $AB \cong CB$

• Reason: This is given directly from the diagram as the two sides $AB$ and $BC$ have the same number of tick marks, indicating they are congruent.
• Reason: Given

Step 4: $\triangle ABD \cong \triangle CBD$

• Reason: At this point, we have two congruent sides $AB \cong CB$, two congruent angles $\angle A \cong \angle C$, and $AD \cong CD$. This is enough to conclude that $\triangle ABD \cong \triangle CBD$ by the SAS (Side-Angle-Side) Congruence Postulate.

Would you like further clarification on any of the reasoning steps? Here are five related questions to deepen your understanding:

1. What other congruence postulates or theorems could be applied if different information was given in this triangle?
2. Why does the midpoint ensure that $AD = CD$?
3. What does the isosceles triangle theorem state, and how does it apply here?
4. How can you prove two triangles are congruent using the Side-Angle-Side (SAS) postulate?
5. What role do base angles play in determining the congruence of triangles?

Tip: When proving triangles congruent, always look for congruent sides and angles based on givens, definitions, and postulates like SAS, SSS, or ASA.