The image appears to contain a geometric problem related to a right-angled triangle ABC. It includes several parts to prove, including triangle congruence and properties of certain lines and intersections.

Here’s a breakdown of the problem:

Given:

• Triangle ABC with a right angle at A.
• M is a point where the angle bisector of angle BAC intersects BC.
• Part (a) asks to prove that triangle ABM is congruent to triangle ACM.
• Part (b) gives that M lies on a line parallel to AC, and it intersects AB at K. The task is to prove triangle congruence by showing that KA = KM and that K is the midpoint of AB.
• Part (c) asks to prove that the sum of sides AB + BC is greater than 2BE.

I will walk through the solution if you’d like more details on each part. Let me know if you'd like to proceed or focus on specific sections of the proof!

Additionally, here are 5 related questions:

1. How can triangle congruence be proven using angle bisectors?
2. What properties of midpoints help in proving triangle congruence?
3. How does the parallelism of lines affect geometric properties in triangles?
4. Why is the intersection point of lines in geometry important for proofs?
5. What theorem can be applied to prove inequalities involving sides of a triangle?

Tip: When dealing with triangle congruence, focus on matching sides and angles systematically, ensuring you apply the correct postulates like SSS, SAS, or ASA.