There are many shape files that have no explicit projection defined for them (i.e., the have no .prj file telling the ArcGIS program what projection the layer is in). For these layers to work properly in ArcMap, it is useful to define the projection explicitly using the instructions below. But remember, all shape files are in some projection, whether it is defined for them or not! This is important to understand before defining a projection for a shape file. You cannot pick just any projection, or the projection you would like it to be in. You have to know what projection it is actually in, and then define it as such. If you do not know what projection the shape file is in, you need to look for metadata or contact the data producer. You can also try guessing, but it is much more efficient to know for sure from the producer of the data.
To define a projection for a shape file, you need to use Arc Toolbox. You will need write permission to the file to do this. The process described below will create a new file with the extension .prj that must go with the rest of the shape file files.
Make sure that the shape file is not currently opened in ArcMap. If it is, remove it from the ArcMap session.
Open Arc Toolbox (Start-ArcGIS-Arc Toolbox)
Go to Data Management Tools - Projections - Define Projection Wizard (shapefiles, geodatabases)
In the Define Projection Wizard dialog box, click on the folder icon to navigate to the folder where your shape file is located.
Add that file (note that you can add mutliple files assuming you are going to define the same projection for all of them - this is very handy some times)
If you get a message that the layer already has a projection defined, and you see that it is the correct projection, just hit cancel, or if it is with other layers that do need definitions created, click on the X to remove it from the list.
If you are proceeding with the definition, click on Next
Click on Select Coordinate System
There are two common methods for selecting a coordinate system:
If you have other files on the computer with the target coordinate system already defined, you can click on the Import button, navigate to the defined shape file and have it use that same coordinate system for the undefined shape file as well
Or you can press Select..., and choose the coordinate system to apply - for the city of Austin data, you would choose Projected Coordinate System - State Plane - NAD 83 (feet) - NAD 83 State Plane Texas Central..., then click on Add.
Press OK
Press Next
Press Finish