Calculating angle using three WGS 84 points

QGIS >= 2.18

New function (after this question was posted) azimuth() has become available, and made this calculation easier.

If we have two point layers, one of which is your samples layer and the other is the specific third point. In the example below they correspond to:

• TargetPoints_4326: sample points. Only unique id ("id") field is required. I have only three ("id"=1,2,3) pink-colored points in this example.
• ObsPoints_4326: your observation points to measure the angle. I have three overlapping points (green color). This layer has to have "EPSG_code" field to define the CRS you want to work, along with "Start" and "End" fields as pointers to the "id" field of the sample points layer.

Then open the attribute table of your observation points layer (e.g. ObsPoints_4326) and create a new field ("angle") with following expression:

degrees(
 azimuth(transform($geometry,'EPSG:4326',"EPSG_code"),
         transform(geometry(get_feature('TargetPoints_4326', 'id', "Start")),
                                                   'EPSG:4326',"EPSG_code"))
-azimuth(transform($geometry,'EPSG:4326',"EPSG_code"), 
         transform(geometry(get_feature('TargetPoints_4326', 'id', "End")),
                                                   'EPSG:4326',"EPSG_code"))
       )

Output ("angle") field is measured anti-clockwise (e.g. first row on the attached picture shows angle= 27.185). If you want clockwise- measurement, please edit the above expression to switch "Start" and "End".

Note: This example used Lambert Conformal Conic (EPSG:3034) which will be good if your points are spread across large area in Europe. If your study area is local, UTM would do.

Despite the @Kazuhito's answer that solves the OP's problem, here is my workflow that also includes the calculation of distance in meters between one point and other two points.

Use case

I have a CSV file with the coordinates of three points in the EPSG:4326 coordinate system. Point 2 is my reference coordinate. I want to find the distance between point 2 and points 1 and 3. I also want to find the angle between the line from points 3 to 2, and the line from point 1 to 2:

Workflow

1. Insert new text delimited layer from CSV file
2. Convert CSV layer to shapefile
3. Use 'Reproject layer' tool in Processing Toolbox to reproject to a suitable Projected Coordinate System
4. Duplicate the reprojected layer twice. In 1 copy, keep only point 2 in the Attribute table, and delete the rest. In the other copy, keep only points 1 and 3.
5. Use the 'Distance to nearest hub (line to hub)' tool in the Processing toolbox. Set source points later to the reprojected points 1 and 3 layer, and target hub layer to reprojected point 2 layer. This should create a new Hub distance layer.
6. In the Attribute table of your Hub distance layer, toggle on the edit mode, and open the Field Calculator, inputting the following
7. Now for each of your lines in the Hub distance layer, you will have an azimuth under the Angle attribute. The absolute difference between the 2 azimuths will give the angle between the 2 lines.

Links to other questions and solutions I used: calculate angle between two lines