how do i cut these angles?

what tool would i use and how would i go about cutting a piece of mdf to lay flush with another piece of mdf at an angle?

Probably the easiest, quickest was is to use a table saw or radial arm saw. You can tilt the blade to whatever angle you need. Downside: You have to fiddle to get the precise angle you're looking for. You should definitely practice with some scrap wood and fit some boards together to make sure you have the correct angle before trying it on the wood that will make up your project.

If the walls of your box are going to be set at some decent angle like 90, 60, 45, etc degrees, then you can also use a router with a bit at half of whatever angle you have (45, 30, 22.5, etc degrees). Some of those numbers are for illustration purposes- I can't specifically remember seing a 22.5 degree router bit. With the correct bit all you have to do is to run the router down the edge of the board. The advantage of the router is that the angle will be exactly what the bit says it is, as long as the router is held flat on the board.

In the long run it's probably better to use a table saw or radial arm saw: 1) You can do multiple cuts quicker, 2) You can vary the angle of the blade to make minor adjustments in the angles. Especially helpful if the box is fitting into a corner with an oddball angle in it.

Unless you have some masochistic streak in you I wouldn't even attempt doing it with a belt sander, hand plane, scroller saw, hand saw, etc.

For braces that aren't very wide you can use a miter box with a hand saw, or a power miter saw. Panels for a box won't fit in one of these.

In this diagram, if side 1 were 15" and side 2 were 14", the hypotenuse side would be 20.5" [the square root of (side one squared plus side two squared)]

If you enter the numbers into the calculator in the link I provided above, using 15 for side one and 14 for side two, the angles of the cut are shown for you.  The angles are calculated at 47 degrees for angle 1, and 43 degrees for angle 2.