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Is this SSS SAS ASA AAS HL or not congruent? And don't assume any measurements

The diagram you've shared shows a quadrilateral, with point \( C \) being a vertex, and \( \overline{CB} \) bisecting \( \angle DCA \) and \( \angle DBA \). Given that line segment \( \overline{CB} \) bisects these two angles, we can infer certain properties about the triangles involved.

### Analyzing the Triangles:

There are two triangles to consider:
1. \( \triangle DCB \)
2. \( \triangle ACB \)

#### What we know:
- \( \overline{CB} \) is a common side between both triangles.
- \( \overline{CB} \) bisects \( \angle DCA \) and \( \angle DBA \), meaning that the angles formed by \( \overline{CB} \) on each side are congruent.

#### Possible Congruence Criteria:
1. **Side-Side-Side (SSS)**: We would need three corresponding sides congruent in both triangles. Since no specific side lengths are given, this does not apply.
2. **Side-Angle-Side (SAS)**: This requires two sides and the included angle to be congruent. Here, we know \( \overline{CB} \) is a common side, and the bisected angles are congruent. If we assume symmetry in the remaining side lengths \( \overline{CD} \) and \( \overline{CA} \), SAS congruence may apply.
3. **Angle-Side-Angle (ASA)**: We need two angles and the included side to be congruent. The bisected angles are congruent, and the shared side \( \overline{CB} \) is common, so ASA is a valid possibility if opposite angles are congruent as well.
4. **Angle-Angle-Side (AAS)**: This applies when two angles and a non-included side are congruent. Since we only have information about the bisected angles and a common side, this criterion cannot be concluded.
5. **Hypotenuse-Leg (HL)**: This is for right triangles, but no indication of right angles is present.

### Conclusion:
Based on the given information, the triangles \( \triangle DCB \) and \( \triangle ACB \) are **congruent by ASA** (Angle-Side-Angle) since we have two congruent angles (from the bisected angles) and a common side \( \overline{CB} \).

Would you like further details or have any questions?

Here are 5 follow-up questions for deeper understanding:
1. What is the Angle-Side-Angle (ASA) criterion for triangle congruence?
2. How does an angle bisector affect the angles in triangles?
3. What other conditions could confirm triangle congruence besides ASA?
4. How can we use the property of a common side in proving congruence?
5. What real-world applications can congruent triangles be found in?

**Tip**: Always look for shared sides or bisected angles when trying to prove congruence between triangles.

Is this SSS, SAS, ASA, AAS, HL or not congruent? And don't assume any measurements.

This problem demonstrates how to prove the congruence of two triangles formed by a quadrilateral using the Angle-Side-Angle (ASA) criterion. Explore the properties of angle bisectors and their role in establishing triangle congruence. Suitable for students in Grades 9-10.

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