Given that BC = 25 cm, AC = 20√2 cm, and angle A = 45 degrees, we can find the other missing angles and side lengths using trigonometry.

Finding angle B:
We know that the sum of the angles in a triangle is 180 degrees. So, we can find angle C first:
Angle C = 180 - (45 + B)
Angle C = 180 - 45 - B
Angle C = 135 - B

Now, we also know that in a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. So, we can use the sine rule to find angle B:
sin(45) = AC / BC
sin(45) = 20√2 / 25
sin(45) = 0.894

Now, we can find angle B using the sine formula:
sin(B) = BC sin(45) / AC
sin(B) = 25 0.894 / 20√2
sin(B) = 1.118

Therefore, angle B is approximately 47.69 degrees.

Finding angle C:
Angle C = 135 - B
Angle C = 135 - 47.69
Angle C = 87.31 degrees

Finding AB:
Use the cosine rule to find AB:
cos(45) = AB / BC
AB = BC cos(45)
AB = 25 cos(45)
AB = 17.68 cm

Therefore, angle B is approximately 47.69 degrees, angle C is approximately 87.31 degrees, and AB is 17.68 cm.