How do you determine the location of the 16 extreme compass points of the contiguous United States? I'll describe my methodology here, which I developed in January 2016 while I was planning my trip around America. My methodology is, I believe, both simple and logical.
First of all, here's a compass rose. Note that 0 degrees (or 360) is at the top, indicating north and 180 degrees indicates south. Each of the 16 compass points, in fact, correspond to a specific bearing.
This table lists the bearings of the four cardinal directions (north, south, east and west) and four ordinal directions:
Direction  Bearing  
North  0  
Northeast  45  
East  90  
Southeast  135  
South  180  
Southwest  235  
West  270  
Northwest  315  
The Basic Approach
To determine the location of each extreme compass point in the U.S. (or any country) you use this same basic approach:

That's the basic approach regardless of which extreme point you're trying to determine, whether it's 0 degrees (the northernmost point), 2 degrees, 45 degrees (the northeasternmost point) or any other point on the compass. However, there's a bit more to it than this, as described below:
1. Determining the Extreme Cardinal Points (i.e., North, South, East, and West)
You may be able to determine the extreme cardinal points of the United States  or any country  quite easily if you simply look at a map. It helps if your map uses the Mercator projection because Mercator projections show true directions. In other words, whatever direction you draw on a Mercator map is the actual, true direction. This is why Mercator maps have been used by ocean navigators for hundreds of years. Google Maps, as one example, use the Mercator projection.
On your map, if the U.S./Canada boundary is a straight line from east to west, it likely uses the Mercator projection. If the U.S./Canada boundary line is curved, it doesn't use the Mercator projection. Along with using a Mercator projection, it also helps if your map shows lines of latitude and longitude, because you can use them as a reference.
You can calculate the extreme cardinal points either graphically (simply by looking at the map) or numerically (by calculating the longitude and latitude values for each point on the perimeter of the United States). To calculate an extreme cardinal point graphically:

You can also calculate this same point numerically, without using any maps or illustrations, if you have access to computer mapping software. To do this:

Either way, graphically or numerically, it’s easy to find an extreme cardinal point, such as the northernmost point or westernmost point of the contiguous United States.
2. Determining the Extreme Ordinal Points
But what about the ordinal directions, like northeasternmost or southwesternmost point? How do you figure out the “southwesternmost” point of the United States? As it turns out, you use the same methodology as above.
In the case of ordinal directions, like "southwesternmost," both directions  south and west in this case  have the same weight or value. For the direction of southwest, “south” is no more (or less) important than “west.” That's because the direction of “southwest” is exactly in between the directions of south and west. The southwestern direction isn’t more (or less) “south” than “west,” just as the northeastern direction isn’t more (or less) “north” than “east.” Both directions have exactly the same weight.
So to determine an extreme ordinal point, such as the southwesternmost point of the United States, you use the same methodology as you used to determine the extreme cardinal direction points described above. To find the southwesternmost point:

You can also do this calculation numerically rather than graphically to determine the extreme ordinal location. In that case, you would determine the latitude and longitude of each point on the perimeter, then you would ADD the x/y coordinate values of each point together. The point with the highest combined value of x and y is the extreme ordinal point. Before adding them, though, you might first have to perform a transformation depending on the direction, since latitude gets smaller as you go south and longitude gets smaller (or more precisely, more negative) as you travel west. Therefore,

So my “point” (heh, heh) is that you can calculate the extreme ordinal points either graphically or numerically, just as with the extreme cardinal directions (north, south, east or west) described above. And you should come up with the same point as being the most extreme regardless of which way you calculate it, either graphically or numerically.
Below: Many people mistakenly believe that San Diego, California is the southwesternmost point of the contiguous United States. Before I left on my trip around America in 2016 and created this website, I Googled "Southwesternmost point of the U.S." and almost every entry claimed it was San Diego. But it's not. To determine the southwesternmost point, 1). Create a line in the Pacific Ocean that's perpendicular to that direction (i.e., bearing 135 degrees, or in other words, from northwest to southeast), then 2). Move that line towards the U.S. until it touches the first point of land. I've done this in this map with the red line. If you create a line bearing 135 through San Diego (blue point and blue line), you'll see that there are many other places in California farther southwest than San Diego, including Santa Barbara and even Big Sur up near Monterey. In fact, the southwesternmost point in the U.S. is an unnamed point near Lompoc, California (red dot). You can use this same approach to calculate any extreme compass point in the U.S. Sorry, San Diego. But congratulations Lompoc! 
3. Determining the Extreme Interordinal Points
You use the same methodology to determine the extreme interordinal points, such as the southsoutheasternmost point or the westsouthwesternmost point. Again, you can do this either graphically or numerically and both ways should yield the exact same location.

But again, you should come up with the exact same interordinal extreme point regardless of whether you’re doing it graphically or numerically.
Conclusion
Using my GIS (geographic information systems) software, I performed these calculations, both numerically and graphically, in January 2016 as I was planning my trip around America. By doing this I was able to determine the 16 extreme compass points of the contiguous United States (i.e., the "Lower 48.") For each of the 16 points, both my numerical calculations and the graphical method yielded the same exact location.
The term "southwesternmost" or "eastnortheasternmost" may be a bit fuzzy or nebulous to some folks. But to me, the southwesternmost or eastnortheasternmost points of the United States are just as clearlydefined, exact and identifiable as the northernmost or westernmost points.
You can use this same method to identify all 16 extreme compass points of any land mass  or 32 points, or even 360 points. But visiting all 360 extreme compass points of the United States would take a lot more time, so on this trip I decided to limit it to just 16!