When you look at the 45-45-90 triangle, label the smaller sides length 1. If they are each 1, then you can use the pythagorean theorem to find the missing hypotenuse.
1^2 +1^2 = c^2 –> c=√2
When you look at the 30-60-90 triangle, label the smallest side length one. When you look at the related 60-60-60 equilateral triangle, all of its sides would be length 2. This now gives you two out of three sides of your 30-60-90 triangle, allowing you to use the pythagorean theorem to find the third side. (Note: the missing side here is not the hypotenuse…)
Now that we know the side lengths of 45-45-90 and 30-60-90 triangles, acknowledge that all 45-45-90 triangles are similar (by AA similarity). So the ratios of their corresponding sides will be proportional. So if you know one of the sides, and want to find the other missing sides, you just set up proportions. Think: Smallest/smallest=medium/medium=largest/largest
smallest/medium (in 1 triangle) = smallest/medium (in the other similar triangle)
largest/medium (in 1 triangle) = largest/medium (in the other similar triangle)
smallest/largest (in 1 triangle) = smallest/largest (in the other similar triangle)