Question

question
determine if the two triangles are necessarily congruent. if so, fill in a flowchart proof to show that they are.
the triangles
necessarily congruent.

Explanation:

Step1: Identify Given Information

In triangle \( KLM \) and triangle \( YAZ \), we have marked angles and sides. Let's check the congruence criteria (ASA, SAS, SSS, AAS).

Looking at the triangles:
- In \( \triangle KLM \): One angle (at \( L \)) and one angle (at \( M \)) with a side between them? Wait, the marked sides: in \( \triangle KLM \), there's a side with one mark, and another side? Wait, the diagram: \( \triangle KLM \) has a marked angle at \( L \), a marked angle at \( M \), and a side with one tick mark, and another side? Wait, maybe it's ASA or AAS. Wait, let's re-examine.

Wait, the triangles: \( \triangle KLM \) and \( \triangle YAZ \). Let's list the corresponding parts:

- Angle at \( L \) (in \( \triangle KLM \)) and angle at \( Y \) (in \( \triangle YAZ \)): are they equal? The marks (the angle symbols) suggest they are equal.
- Angle at \( M \) (in \( \triangle KLM \)) and angle at \( Z \) (in \( \triangle YAZ \)): the marks suggest they are equal.
- The side between the angles? Wait, no, the side with the tick mark: in \( \triangle KLM \), the side from \( M \) to the middle? Wait, maybe it's AAS. Wait, AAS: two angles and a non-included side. Or ASA: two angles and included side.

Wait, let's check the sides: In \( \triangle KLM \), there's a side with one tick mark, and in \( \triangle YAZ \), a side with one tick mark. Wait, maybe the sides are equal. Wait, the key is: if two angles and a side (either included or non-included) are equal, then triangles are congruent.

Wait, let's see: \( \angle L = \angle Y \) (angle marks), \( \angle M = \angle Z \) (angle marks), and the side between them? Wait, no, the side with the tick mark: in \( \triangle KLM \), the side from \( M \) to the side with the tick mark, and in \( \triangle YAZ \), the side from \( Z \) to the side with the tick mark? Wait, maybe it's AAS. Wait, AAS: two angles and a non-included side. If two angles are equal, and a side (not between them) is equal, then triangles are congruent. But here, maybe the side is included? Wait, no, let's think again.

Wait, the triangles: Let's label the vertices. \( \triangle KLM \): vertices \( K, L, M \). \( \triangle YAZ \): vertices \( Y, A, Z \).

Looking at the angles:
- \( \angle L \cong \angle Y \) (angle marks)
- \( \angle M \cong \angle Z \) (angle marks)
- The side \( LM \) and \( YZ \)? Wait, no, the side with the tick mark: in \( \triangle KLM \), the side from \( M \) to the side with one tick, and in \( \triangle YAZ \), the side from \( Z \) to the side with one tick? Wait, maybe the side \( KM \) and \( YZ \)? No, maybe it's the side between the angles? Wait, no, if two angles are equal, and the side opposite one of the angles is equal, then AAS. Wait, but let's confirm.

Wait, the problem is to determine if they are necessarily congruent. Let's recall congruence criteria:

- ASA: Two angles and included side.
- AAS: Two angles and non-included side.
- SSS: Three sides.
- SAS: Two sides and included angle.

In this case, we have two angles equal ( \( \angle L = \angle Y \), \( \angle M = \angle Z \) ) and a side equal (the side with the tick mark, which is a non-included side? Wait, no, if two angles are equal, and a side (either included or not) is equal, then triangles are congruent. Wait, AAS: two angles and a non-included side. So if \( \angle L = \angle Y \), \( \angle M = \angle Z \), and \( LM = YZ \) (the side with the tick mark), then by AAS, triangles are congruent. Wait, but maybe the side is included. Wait, no, the included side would be between the two angles. If \( \angle L \) and \( \angle M \) are in \( \triangle KLM \), the included side is \( LM \). If \( \angle Y \) and \( \angle Z \) are in \( \triangle YAZ \), the included side is \( YZ \). If \( LM = YZ \), then ASA (two angles and included side) would apply.

Wait, the diagram: in \( \triangle KLM \), the side between \( \angle L \) and \( \angle M \) is \( LM \), and in \( \triangle YAZ \), the side between \( \angle Y \) and \( \angle Z \) is \( YZ \). If \( \angle L = \angle Y \), \( \angle M = \angle Z \), and \( LM = YZ \), then by ASA, \( \triangle KLM \cong \triangle YAZ \). Therefore, the triangles are necessarily congruent.

Step2: Conclusion

Since two angles and the included side are equal (ASA congruence criterion), the triangles are necessarily congruent.

Answer:

The triangles are necessarily congruent.