Is there any way to find angle $v$ in the trapezoid below?
I only know sides $a$ and $b$. My own idea was to cut the trapezoid into two triangles and form two equations like so:-
And then form the equations $$45^\circ+v+w=180^\circ$$ and $$q+90^\circ+45^\circ=180$$
But then I don't utilize the known side lengths, and I'm not even sure this is correct. Can someone help me out?
A way for you to decide whether you can determine this (in case this situation comes up in a setting where you need to answer your own question) is to draw out the picture (like you have already done). Instead of drawing the entire thing though, only draw part of the shape, in this case yoou have 3 sides in which you have angles defined. You might need to do a few cases since the bottom and left side lengths are undefined. The way I did this was to make one picture with the left side short, one medium, and one long. Next grab a ruler (or anything you can use determine congruent lengths). Since you stated that you knew the value for b then it is a defined (or constant length). So choose a length that is sufficiently long enough to reach from the endpoint of the top side to the end point of the longest bottom side. Continue using the same length and connect the top to the bottom. What do you notice? Did you have to adjust the length of the bottom to make this work? Was that alllowed by the parameters you set? You should have had to adjust the base lengths for each "small," "medium," or "large" side length to be able to keep your b constant. In changing the length of the base, you also change the angle between the base and the side of length b.
When doing this type of problem solving, you are using concrete examples to get a visual understanding of the concept/ rules that make up the quetion itself. You can apply this type of thinking to just about any geometric problem where it is possible to draw a diagram. Just make sure you keep your constants (what you have defined to be a certain measurement) the same in each situation and only change that which you have not given any constraints to.
In the picture, you will see my cases 1, 2 and 3. All have set values for a and b but obviously have completely different measures of angle V.