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The image shows a mathematical problem written in Sinhala, with a focus on proving the congruence of two triangles, \( \triangle ABX \) and \( \triangle BOY \), as highlighted in the green section. 

### Explanation:
1. The problem asks for proof that \( \triangle ABX \equiv \triangle BOY \). To prove triangle congruence, one typically uses criteria such as:
   - **SSS (Side-Side-Side)**: All three sides of one triangle are equal to the corresponding sides of the other.
   - **SAS (Side-Angle-Side)**: Two sides and the included angle of one triangle are equal to those of the other.
   - **ASA (Angle-Side-Angle)**: Two angles and the included side of one triangle are equal to those of the other.
   - **RHS (Right-Angle Hypotenuse Side)**: Applicable for right-angled triangles, where the hypotenuse and one side are equal.

In your case, it seems like you will need to identify matching sides and angles based on the given information. However, without the specific geometry details (such as coordinates or side lengths), I can suggest a method based on possible congruence criteria.

### How to Approach:
1. Identify whether the problem gives any side lengths, angles, or other geometrical properties that match between \( \triangle ABX \) and \( \triangle BOY \).
2. Use the appropriate congruence criterion (like SSS, SAS, ASA, or RHS) to prove the congruence.

Would you like me to go over specific criteria or details for solving triangle congruence problems?

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### Related Questions:
1. What are the properties of congruent triangles?
2. How can SSS, SAS, or ASA be applied to prove triangle congruence?
3. Are there any additional properties such as midpoint or parallel sides mentioned in the problem?
4. How do you prove triangle congruence using transformations (reflection, rotation, translation)?
5. What role do angles and sides play in determining triangle congruence?

### Tip:
When proving triangles congruent, it's helpful to check for shared sides or angles, which often provide quick evidence for the proof.

Prove that triangle ABX is congruent to triangle BOY.

Learn how to prove that triangle ABX is congruent to triangle BOY using triangle congruence criteria like SSS, SAS, ASA, or RHS. This problem covers essential geometry concepts and is ideal for students in Grades 8-10.

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