Question

based on the given diagram, 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 Congruent Parts

From the diagram, we see that \( \overline{BE} \cong \overline{BD} \) (marked with tick marks) and \( \angle AEB \cong \angle CDB \) (marked with arc marks). Also, \( \angle ABE \) and \( \angle CBD \) are the same angle (common angle), so \( \angle ABE \cong \angle CBD \).

Step2: Choose Congruence Criterion

We can use the ASA (Angle - Side - Angle) or SAS (Side - Angle - Side) criterion. Let's check:

• Side: \( \overline{BE} \cong \overline{BD} \) (given by tick marks).
• Angle: \( \angle AEB \cong \angle CDB \) (given by arc marks).
• Angle: \( \angle ABE \cong \angle CBD \) (common angle).

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

Step3: Fill the Flowchart Statements and Reasons

• First statement (side): \( \overline{BE} \cong \overline{BD} \), Reason: Given (marked congruent segments).
• Second statement (angle): \( \angle AEB \cong \angle CDB \), Reason: Given (marked congruent angles).
• Third statement (angle): \( \angle ABE \cong \angle CBD \), Reason: Common angle (reflexive property of congruence for angles).
• Final statement: \( \triangle ABE \cong \triangle CBD \), Reason: ASA (Angle - Side - Angle) Congruence Criterion.

Answer:

To prove \( \triangle ABE \cong \triangle CBD \):

1. Statement: \( \boldsymbol{\overline{BE} \cong \overline{BD}} \), Reason: Given (marked congruent segments).
2. Statement: \( \boldsymbol{\angle AEB \cong \angle CDB} \), Reason: Given (marked congruent angles).
3. Statement: \( \boldsymbol{\angle ABE \cong \angle CBD} \), Reason: Common angle (reflexive property).
4. Statement: \( \boldsymbol{\triangle ABE \cong \triangle CBD} \), Reason: ASA (Angle - Side - Angle) Congruence.