https://file.aiquickdraw.com/mathbot/file/1729703811108l0rzy6eq.jpg

The diagram provided shows two triangles, \(\triangle ABD\) and \(\triangle CBD\), sharing a common side \(BD\), with \(\overline{AB} = \overline{BC}\) and \(\overline{BD}\) being a common side for both triangles. Additionally, \(\angle ABD = \angle CBD = 90^\circ\) as they form right angles at point \(B\).

To prove that these triangles are congruent, we can use the **SAS (Side-Angle-Side) Postulate**. Here’s why:

1. **Side**: \(\overline{BD}\) is shared by both triangles.
2. **Angle**: \(\angle ABD = \angle CBD = 90^\circ\), both are right angles.
3. **Side**: \(\overline{AB} = \overline{BC}\), as given by the tick marks showing they are equal in length.

Since we have two sides and the included angle congruent, the **SAS (Side-Angle-Side)** postulate applies.

The correct answer is **SAS**.

Would you like more details, or do you have any further questions?

Here are 5 related questions you can explore:
1. What other methods can be used to prove triangle congruence?
2. Can the AAS (Angle-Angle-Side) postulate apply here?
3. Why is SSA (Side-Side-Angle) not a valid congruence postulate?
4. How does the HL (Hypotenuse-Leg) theorem apply to right triangles?
5. How do you differentiate between congruent and similar triangles?

**Tip**: Always check for shared sides or angles when proving triangle congruence, as these often provide key clues.

Determine which postulate can be used to prove that the triangles are congruent (based on the diagram where triangles ABD and CBD share side BD, and AB = BC).

Learn how to prove the congruence of triangles ABD and CBD using the SAS (Side-Angle-Side) postulate. This problem explores key concepts in triangle congruence and geometric properties, suitable for students in Grades 8-10.

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