Geometry

Question Analysis:
This question aims to test the understanding of multiple triangle congruence theorems by presenting a scenario with two right - angled triangles.
The main focus is to match the given side - length and angle information with the requirements of various congruence theorems.

Key Concept Explanation:
- SSS (Side - Side - Side): Triangles are congruent when the lengths of all three sides of one triangle are equal to the lengths of the corresponding sides of the other triangle.
- SAS (Side - Angle - Side): Requires two sides and the included angle of one triangle to be congruent to the corresponding parts of the other triangle.
- HL (Hypotenuse - Leg): Specifically for right - triangles, if the hypotenuse and one leg of one right - triangle are congruent to the hypotenuse and one leg of another right - triangle, the triangles are congruent.
- ASA (Angle - Side - Angle): Two angles and the included side of one triangle must be congruent to the corresponding parts of the other triangle for congruence.

Step - by - Step Solution:
1. Given that , , , and .
2. For SSS:
  - Since all three sides of (, , ) are congruent to the corresponding three sides of (, , ), the SSS theorem can be used to prove .
3. For SAS:
  - We have two sides

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