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Which dataset determines the coordinate system of my data frame?
The first layer added to the data frame defines its coordinate system. This is true whether the data is projected or geographic. For example, if the first layer added contains a Lambert Conformal Conic projected coordinate system, all other layers will project on the fly to match this. Similarly, if the first layer added to the data frame contains data that uses a WGS84 geographic coordinate system, all other layers will adjust to match this. Even data that uses a projected coordinate system will unproject on the fly.
To force the data frame to use a coordinate system of any subsequently-added layer, click Data Frame Properties > Coordinate System, and select a coordinate system from the Layers folder.