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 to 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 congruency ticks). And $\angle ABE$ and $\angle CBD$ are the same angle (common angle), so $\angle ABE \cong \angle CBD$.

Step2: Apply ASA Congruence Criterion

To prove $\triangle ABE \cong \triangle CBD$, we use the Angle - Side - Angle (ASA) congruence postulate.

• Angle: $\angle A \cong \angle C$ (given by the congruency marks on the angles).
• Side: $\overline{AB} \cong \overline{CB}$ (given).
• Angle: $\angle ABE \cong \angle CBD$ (common angle, reflexive property of congruence).

By ASA, if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. So $\triangle ABE \cong \triangle CBD$ by ASA.

Answer:

To fill the flowchart:

1. First statement (angle): $\angle A \cong \angle C$ (Reason: Given, from diagram marks)
2. Second statement (side): $\overline{AB} \cong \overline{CB}$ (Reason: Given)
3. Third statement (angle): $\angle ABE \cong \angle CBD$ (Reason: Common angle, reflexive property)
4. Final statement: $\triangle ABE \cong \triangle CBD$ (Reason: ASA Congruence Postulate)