Question

given \\(\overline{ab} \cong \overline{cb}\\), prove \\(\triangle abe \cong \triangle cbd\\) by filling out the flowchart below.
for each box, choose a statement format from the dropdown menu. you will then be able
change the letters to match the diagram for this problem.

Explanation:

Step1: Identify Given Information

We know $\overline{AB} \cong \overline{CB}$ (given). Also, from the diagram, $\angle A \cong \angle C$ (marked with the same symbol).

Step2: Identify Vertical Angles or Common Angle? Wait, Let's Check the Triangles.

Looking at $\triangle ABE$ and $\triangle CBD$, we have $\angle ABE$ and $\angle CBD$? Wait, no, maybe the included angle? Wait, actually, let's see the angles. Wait, $\angle AEB$ and $\angle CDB$? No, maybe the vertical angles? Wait, no, let's re - examine. Wait, the triangles share $\angle B$? Wait, no, $\overline{AB} \cong \overline{CB}$, $\angle A \cong \angle C$, and what about the other angle or side? Wait, maybe $\angle ABE \cong \angle CBD$? No, wait, actually, the vertical angles? Wait, no, let's think again.

Wait, the correct approach: We have $\overline{AB} \cong \overline{CB}$ (given), $\angle A \cong \angle C$ (marked), and $\angle ABE \cong \angle CBD$? No, wait, actually, $\angle AEB$ and $\angle CDB$? No, maybe the vertical angles formed by the intersection of the two lines. Wait, the two lines $AE$ and $CD$ intersect? No, looking at the diagram, the two triangles are $\triangle ABE$ and $\triangle CBD$. So, $\overline{AB} \cong \overline{CB}$ (given), $\angle A \cong \angle C$ (marked), and $\angle ABE \cong \angle CBD$? Wait, no, $\angle ABE$ and $\angle CBD$ are the same angle? Wait, no, $\angle B$ is common? Wait, $\angle ABE$ and $\angle CBD$: if we consider the angle at $B$, yes, $\angle ABE = \angle CBD$ (common angle? Wait, no, $\angle ABE$ and $\angle CBD$ are the same angle because they are the angle at vertex $B$ for both triangles. Wait, no, actually, $\angle ABE$ and $\angle CBD$: let's see, the angle at $B$ for $\triangle ABE$ is $\angle ABE$, and for $\triangle CBD$ is $\angle CBD$, and they are equal because they are the same angle (common angle).

Wait, now, using ASA (Angle - Side - Angle) congruence criterion. We have:

• $\overline{AB} \cong \overline{CB}$ (given)
• $\angle A \cong \angle C$ (marked)
• $\angle ABE \cong \angle CBD$ (common angle or vertical angles? Wait, no, it's the same angle at $B$, so $\angle ABE=\angle CBD$ (reflexive property? No, it's the same angle, so congruent).

So, by ASA (Angle - Side - Angle) congruence postulate, $\triangle ABE \cong \triangle CBD$.

Answer:

To prove $\triangle ABE \cong \triangle CBD$:

1. Statement 1: $\overline{AB} \cong \overline{CB}$; Reason: Given.
2. Statement 2: $\angle A \cong \angle C$; Reason: Marked in the diagram (congruent angles).
3. Statement 3: $\angle ABE \cong \angle CBD$; Reason: Common angle (or reflexive property of angle congruence).
4. Statement 4: $\triangle ABE \cong \triangle CBD$; Reason: ASA (Angle - Side - Angle) Congruence Postulate (since we have two angles and the included side congruent: $\angle A \cong \angle C$, $\overline{AB} \cong \overline{CB}$, $\angle ABE \cong \angle CBD$).