To make this problem easier, we will split it into two triangles. First, we need to work on the blue triangle to find the hypotenuse. This is a shared edge with the red triangle. Then we can work on the red triangle

now we can use SOH CAH TOA to solve for the missing angle

Let's set this up as a SOH CAH TOA. I will draw a little guy at the angle I am trying to find, then label the sides with respect to him (and the angle):

Now, as we have Opp and Hyp, we select the SOH:

Plug in the values and solve:

The area is given by finding the area of the red and the blue triangles.
First, we need the height of the red triangle, let's use Pythagoras to solve this.

the area of a triangle is (0.5)*base*height

This is a simple application of Pythagoras theorem. Ok, so we have to reverse it to find one of the "legs" but its simple enough!

Ok, so time to solve the setup from part a

I think the hardest part of this question is understanding what they are trying to say. To cover our backs let's state a general Pythagoras theorem, and then make it relevant to the question by substituting all the values that we can: