1. Chapter 4
Angles and Directions

2. 4.2: Reference Direction for Vertical Angles:
Vertical angles: angle which used in slope distance corrections or in height determination
Vertical angle are referenced to the:
(1) the horizon by plus (up) or minus (down) angles.
(2) zenith.
(3) nadir.
- Zenith and nadir are terms describing points on a celestial sphere .

3. 4.3: Meridians:
Meridians: a line on the mean surface of the earth joining the north and south poles.
geographic meridian: is the line formed by the intersection with the earth’s surface of a plane that includes the earth’s axis of rotation.
Magnetic meridians: are parallel to the directions taken by freely moving magnetized needles, as in a compass.
Grid meridians: are lines that are parallel to a grid reference meridian (central meridian).

4. 4.4: Horizontal Angles:
Horizontal angles are usually measured with a theodolite or total station.
For all closed polygons of n sides, the sum of the interior angles equal to (n-2)180ْ .
For all closed polygons of n sides, the sum of the exterior angles equal to (n+2)180ْ .
Example: What is the sum of the interior angles for a five-sides polygon?
(5-2)180 ْ =
(3)(180) = 540ْ

5. 4.5: Azimuths:
Azimuth: is the direction of a line as given by an angle measured clockwise (usually) from the north end of a meridian.
Azimuths range in magnitude from 0ْ to 360ْ .
Azimuth
Line
52ْ
0-1
121ْ
0-2
202ْ
0-3
325ْ
0-4

6. 4.6: Bearings:
Bearing: is the direction of a line as given by the acute angle between the line and a meridian.
Measured clockwise or counterclockwise from the north or south end of the meridian.
Bearing is always accompanied by letters that locate the quadrant in which the line falls (NE, NW, SE, or SW).

7. Bearing
Line
N58ْ00’E
0-1
S25ْ00’E
0-2
S28ْ00’W
0-3
N10ْ00’W
0-4

8. 4.7: Relationships Between Bearings and Azimuths:
To convert from azimuths to bearings by using this table:
Numerical value
quadrant letters
quadrant
bearing = azimuth NE
From 0ْ to 90ْ
bearing= 180ْ - azimuth SE
From 90ْ to 180ْ
bearing= azimuth – 180ْ SW
From 180ْ to 270ْ
bearing= 360ْ - azimuth NW
From 270ْ to 360ْ

9. To convert from bearings to azimuths by using this relationships:
NE quadrant azimuth = bearing
SE quadrant azimuth = 180ْ – bearing
SW quadrant azimuth = 180ْ – bearing
NW quadrant azimuth = 360ْ – bearing
Example: convert :
200ْ 58’ = S20ْ 58’W
N2ْ 21’W = 357ْ 39’

10. 4.8: Reverse Directions:
It can be said that every line has two direction.
Forward direction is direction that oriented in the direction of fieldwork or computation staging.
Back direction is direction that oriented in the reverse of the direction fieldwork or computation staging.

11. To reverse a bearing … Reverse the direction letters.
Example:
Bearing
Line
N 60ْ 00’ E AB
S 60ْ 00’ W BA

12. To reverse an azimuth …. Add 180ْ to the original direction.
Example:
Azimuth
Line
128ْ 00’ AB
308ْ 00’ BA

13. 4.9: Azimuth Computations:
START
Az AB = 330ْ 00’ (Given)
- 180ْ
Az BA = 150ْ 00’
+<B ْ 28’
Az BC = 270ْ 28’

14. Az BC = 270ْ 28’
- 180ْ
Az CB = 90ْ 28’
+<C = 118ْ 37’
Az CD = 209ْ 05’
Az CD = 209ْ 05’
- 180ْ
Az DC = 29ْ 05’
+<D ْ 22’
Az DE = 134ْ 27’

15. Az DE = 134ْ 27’
+ 180ْ
Az ED = 314ْ 27’
+<E ْ 28’
Az EA = 422ْ 55’
- 360ْ
Az EA = 62ْ 55’
Finish
Check
Az EA = 62ْ 55’
+ 180ْ
Az AE = 242ْ 55’
+<A ْ 05’
Az AB = 330ْ 00’

16. 4.10: Bearing Computation:
Line BC:
(?)= 180 – (120ْ 28’- 30ْ )
(?)= 89ْ 32’ in N.W quad.
i.e., N 89ْ 32’ W
Line CD:
(?)= 118ْ 37’- 89ْ 32’
(?)= 29ْ 05’ in S.W quad.
i.e., S 29ْ 05’ W

17. Line DE:
(?)= 180ْ – (105ْ 22’+29ْ 05’)
(?)= 45ْ 33’ in S.E. quad.
i.e., S 45ْ 33’ E
Line EA:
(?)= 108ْ 28’- 45ْ 33’
(?)= 62ْ 55’ in N.E. quad.
i.e., N 62ْ 55’ E

18. Line AB:
(?)= 180ْ – (62ْ 55’+ 87ْ 05’)
(?)= 30ْ 00’ in N.W. quad.
i.e., N 30ْ 00’ W
CHECK (Line AB was S 30ْ 00’ E )

19. 4.12: Magnetic Direction:
Magnetic Direction is the horizontal angle between magnetic north and geographic north.
Isogonic chart is line joining points of the earth surface having equal magnetic declination.
Geographic Bearing of survey Line = 15ْ 30’+ 10ْ 30’
= 26ْ 00`