The right triangle with one angle measuring 30° must have the other acute angle measuring 60° (because it's a right triangle). This is a 30-60-90 triangle. In a 30-60-90 triangle, the hypotenuse is twice the length of the side opposite the 30° angle.

Let "a" be the length of the side opposite the 30° angle and "c" be the length of the hypotenuse. Then, the hypotenuse (c) = 2a.
We are given the length of side a as 4 cm:

c = 2 * 4 cm
c = 8 cm

Since sin(β) = a/c and cos(γ) = a/c, we can conclude that sin(β) = cos(γ). This implies β and γ are complementary angles, which means:

β = 90° - γ

Given that A = 90°, we have a 45-45-90 right triangle.

In a 30-60-90 triangle, the hypotenuse (c) is twice the length of the side opposite the 30° angle (a). The side opposite the 60° angle (b) is √3 times the length of the side opposite the 30° angle:

b = a√3

We are given the side length opposite the 60° angle as 14 cm:

14 cm = a√3

Divide both sides by √3:

a = 14/√3 cm

Now we can find the length of the hypotenuse:

c = 2a = 2(14/√3) cm = 28/√3 cm

Since triangle LMN is a right triangle with angle M = 45°, angle N must be 45° as well (because angle L is a right angle). The triangle LMN is, therefore, a 45-45-90 triangle. In a 45-45-90 triangle, the sides opposite both of the 45° angles are congruent, and the hypotenuse is √2 times the length of each leg.

In this case, the side LN (opposite angle N) is 9 cm. Since the triangle is a 45-45-90 triangle, the side opposite angle M (MN) must also be 9 cm.

In a 30-60-90 right triangle, the side opposite the 30° angle is half the length of the hypotenuse. Thus, the ratio of the side opposite the 30° angle to the hypotenuse is 1/2.