https://file.aiquickdraw.com/mathbot/file/17297793193358l55ezxy.jpg

This problem asks you to determine why the two triangles \( \triangle NOM \) and \( \triangle ZOY \) are congruent. Let's break it down step by step:

### Why are the two triangles congruent?
From the image, the two triangles share congruent angles and side lengths. Specifically:
1. \( \angle NOM \cong \angle ZOY \) because they are vertically opposite angles.
2. \( \angle NMO \cong \angle ZYO \), as they both have a right angle.
3. The segment \( MO \cong YO \), which is marked as congruent between the triangles.

This suggests that the triangles are congruent by **ASA** (Angle-Side-Angle) congruence criterion.

### What is the congruence statement?
The correct congruence statement would be:
\[ \triangle NOM \cong \triangle ZOY \]

### List all other corresponding parts of the triangles that are congruent:
1. \( \angle N \cong \angle Z \)
2. \( \angle M \cong \angle Y \)
3. \( \overline{MO} \cong \overline{YO} \)

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. How can you prove the Angle-Side-Angle (ASA) congruence theorem?
2. What other congruence theorems exist besides ASA?
3. Why are vertically opposite angles always congruent?
4. Can the congruence of these triangles be proved using a different theorem?
5. What role does the right angle play in proving congruence here?

**Tip:** When dealing with congruent triangles, always identify corresponding parts carefully to ensure the congruence statement is accurate.

Tell why the two triangles are congruent. Give the congruence statement. Then list all other corresponding parts of the triangles that are congruent.

Understand why triangles NOM and ZOY are congruent using the ASA (Angle-Side-Angle) theorem. This detailed solution explains congruent angles, sides, and how vertically opposite angles are used in the proof. Suitable for students in Grades 8-10.

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