4. Angles from Directions - Traverse-based

a. Angles from Bearings

Given bearings going counter-clockwise 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
	BrngRU = S 15°24'47" E AngleR = 71°43'30"+15°24'47" = 87°08'17"
At point S
	BrngSR = N 71°43'30" E AngleS = 108°00'00"-(35°25'40"+71°43'30")              =72°50'50"
At point T
	BrngTS = N 35°25'40" E AngleT = 72°48'15"+35°25'40"             = 108°13'55"
At point U
	BrngUT = S 72°48'15" E AngleU = (180°00'00"-72°48'15")+15°00'35"             = 122°12'20"
At point V
	BrngVU = S 15°00'55" W AngleV = 88°25'02"-15°00'35"             = 73°24'27"
At point W
	BrngWV = N 88°25'02" W AngleW = 360°00'00'-(88°25'02"+15°24'47")             = 256°10'11"

Math check

Six sided polygon: Σ(Int angles) = (n-2) x 180° = (6-2) 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. Because the difference falls outside of the ±180°00'00" deflection angle range, add 360°00'00" to it: The is the same result if you added 360°00'00" to the outgoing azimuth before subtracting the incoming azimuth.

Math check:

For a non-crossing loop traverse, the deflection angle sum should be ±360°