4. Angles from Directions  Traversebased
a. Angles from Bearings
Given bearings going counterclockwise around a loop traverse, compute the interior angles at each point. Perfom a math check at completion.
It doesn't matter where to start or in what order or direction to compute the interior angles.
Drawing sketches helps visualize how to combine the bearings at a point to compute the angle.
At point R  
Brng_{RU} = S 15°24'47" E Angle_{R} = 71°43'30"+15°24'47" = 87°08'17" 

At point S  
Brng_{SR} = N 71°43'30" E Angle_{S} = 108°00'00"(35°25'40"+71°43'30") =72°50'50" 

At point T  
Brng_{TS }= N 35°25'40" E Angle_{T} = 72°48'15"+35°25'40" = 108°13'55"


At point U  
Brng_{UT }= S 72°48'15" E Angle_{U} = (180°00'00"72°48'15")+15°00'35" = 122°12'20" 

At point V  
Brng_{VU }= S 15°00'55" W Angle_{V} = 88°25'02"15°00'35" = 73°24'27" 

At point W  
Brng_{WV }= N 88°25'02" W Angle_{W} = 360°00'00'(88°25'02"+15°24'47") = 256°10'11" 
Math check
Six sided polygon: Σ(Int angles) = (n2) x 180° = (62) x 180° = 720°
Σ(Int angles) = 87°08'17"+72°50'50"+108°13'55"+122°12'20"+73°24'27"+256°10'10" = 720°00'00" check
b. Deflection Angles from Azimuths
(1) Process
Because a deflection angle is the angle from the projection of the previous line to the next line, it is the difference between the azimuths of the two lines. For example, in the diagram below the deflection angle Y is the difference between the outgoing azimuth (Az_{YZ}) and the incoming azimuth (Az_{XY}).
defl ang_{Y} = Az_{YZ}Az_{XY}
By subtracting the incoming azimuth from the outgoing azimuth, the correct mathematical sign is returned for the deflection angle. In the previous diagram, the deflection angle would be to the left. In the following diagram:
the deflection angle is to the right.
Whether you remember to subtract the incoming from the outgoing azimuth or not, the important thing is that the deflection angle is the difference between the azimuths. As in many survey computations, a properly drawn sketch is extremely beneficial.
(2) Example Azimuths to Deflection Angles
Given azimuths going clockwise around a loop traverse, compute the deflection angles in the same travel direction. Perfom a math check at completion
At point L  
At point M  
At point N  
At pointr O  
From the sketch, the deflection angle should be to the right so the difference should be positive. 
Math check:
For a noncrossing loop traverse, the deflection angle sum should be ±360°