1. Definition
A horizontal angle is measured perpendicular to gravity. Three points are involved:

Backsight (BS)

Foresight; FS

At
The angle is between two vertical planes; one contains the BS and At points, the other contains the FS and At. The At point is generally occupied by a surveying instrument, Figure G1.
Figure G1 
If we look down on the horizontal plane with the three points, Figure G2, we see there are two horizontal angles between the vertical planes: HAng_{1} and HAng_{2}.
Figure G2 
This causes an ambiguous situation unless we can specifically identify which angle we mean. Recall that the four parts of an angle are:

Start

Direction

Magnitude

End
HAng_{1} and HAng_{2} have the same start (BS) and end (FS) but different direction and magnitude. All four parts must be clearly defined. Merely including magnitude without a direction doesn’t resolve the ambiguity. For example, a 135° angle from BS to FS could still be interpreted two ways, Figure G3.
Figure G3 
Similarly, expressing a direction, to the right (clockwise) or left (counterclockwise), without a magnitude is also unclear, Figure G4.
Figure G4 
There are a few different ways to specify horizontal angles which help minimize confusing interpretations.
2. Types of Horizontal Angles
a. Interior/Exterior
On a closed noncrossing polygon, Figure G5, horizontal angles can be either interior (red) or exterior (blue).
Figure G5 
The geometric condition for a noncrossing polygon is Equation (G1).
Equation (G1) 
Interior and exterior don’t make sense if the polygon is open, Figure G6(a), or if the polygon crosses itself, Figure G6(b).
a. Open Polygon 
b. Crossing Polygon 
Figure G6 
In the latter case, the polygon turns itself insideout and Equation (E1) does not apply. Open and crossing polygons need some other way to express an angle.
b. Right/Left
Right or left is the rotational direction from the BS point to the FS point. Consider standing on the At point, looking at the BS point: an angle to the right means you physically turn your body to the right to see the FS point. An angle right is clockwise, left counterclockwise, Figure G7.
Figure G7 
Figure G8(a) shows a closed noncrossing polygon with interior angles to the right; Figure G8(b) to the left.
a. Angles Right 
b. Angles Left 
Figure G8 
The dashed lines in Figure G8 show the angle measurement sequence. In Figure G8(a), interior angles to the right measurement progresses counterclockwise around the polygon.
We can also have exterior angles to the right, Figure G9(a), and to the left, Figure G9(b).
a. Angles Right 
b. Angles Left 
Figure G9 
Right or left help with angle definitions on the open and crossing polygons. In Figure G10, progressing in the direction of the dashed line, the open polygon has angles to the right.
Figure G10 
Figure G11 shows a crossing polygon using angles left progressing in the direction of the dashed line.

Figure G11 
Note how the angles change from “interior” to “exterior” although there is no consistent polygon interior.
c. Deflection Angles
A deflection angle is how much the next line deflects from an extension of the previous line. It consists of two parts: (1) magnitude and (2) direction. For example, in Figure G12:
Line BC deflects from line AB 45° to the right, written as 45°R.
Line CD deflects from line BC 30°L.
Figure G12 
A deflection angle ranges from 0° (no deflection) to 180° (going back along the preceding line).
The mathematical angle condition for a closed noncrossing polygon, Figure G13, is:
Equation (G2) 
Right deflection angles are positive (+), left are negative ().
Figure G13 
Equation (G2) is also the mathematical condition for a closed polygon which crosses itself an even number of times. For an odd number of crossings, Figure G14, the mathematical condition is:
Equation (G3) 
Figure G14 
How does reversing travel direction around a polygon change the deflection angles? Figure G15 shows that the deflection angle value is the same, but its direction is reversed.
a. Clockwise Travel 
b. Counterclockwise Travel 
Figure G15 
3. Example Problem
Given this polygon with combination of angle types:
Determine
a. Deflection angle at A from D to B
Defl angle A = 180°00’  113°20’ = 66°40’ = 66°40’ R
b. Angle left at B from A to C
Int angle B = 180°00’  92°42’ = 87°18’
c. Deflection angle at D from A to C
Need interior angle at D to compute the deflection angle.
To compute interior angle at D, subtract the sum of the other interior angles from (n2)x180°.
Need interior angle at C.
Int angle C = 360°00’  259°35’ = 100°25’
Int angle D = (42)x180°00’  (113°20’ + 87°18’ + 100°25’) = 360°00’  301°03’ = 58°57’
Defl angle D = 180°00’  58°57 = 121°03’ = 120°03’ L