You will probably remember learning about linear equations, and being told that if you have n unknown variables, you need (at least) n independent equations for them. So you first want to identify all the variables. In example 1, that's x, y, z, r, and s. So n = 5, and you need to find 5 equations that relate these variables. Then you can solve this system either by substitution or linear combinations.

Geometry is chock full of relationships, so it should not be too hard to find all these equations. (One thing I was confused by - what are the symbols you have written on the problems - the triangle on the lines in example 1, and the arrows in example 2?)

You (should) know supplementary angles sum to 180 deg, yes? So there's a bunch of equations for you right off the bat:

s + r = 180

(2x) + (2x + y) = 180

(2x + y) + z = 180

You also know that

(2x) + (s) = 180

(2x + y) + r = 180

So there's 5 equations right off the bat, without even breaking a sweat.

If you know that the two horizontal lines are parallel, then you can make other equations from the relationships between the groups of angles formed by the vertical line and the two parallel lines, and you can also use vertical angle identities as well.

For the second problem, is any information given about the type of shape this is? Are any of the line pairs parallel? The only thing I can tell for sure without making any assumptions is that the 4 angles should sum to 360 deg.

If either of the line pairs are parallel, you'll be able to use properties of the diagonals to relate opposite angles.