Figure 15-15.-Chain of single triangles.high-order triangulation network to establish controlacross the United States.TYPES OF TRIANGULATION NETWORKSIn triangulation there are three types of triangulationnetworks (or nets). They are the chain of singletriangles, chain of polygons, and chain of quadri-laterals.Chain of Single TrianglesThe simplest triangulation system is the chain ofsingle triangles shown in figure 15-15. Suppose AB isthe base line and measures 780.00 feet in length.Suppose, also, that angle A (that is, the observed angleBAC) measures 98°54´ and that angle ABC measures32°42´. (In actual practice you will use more precisevalues than these; we are using rough values to simplifythe explanation.) Subtracting the sum of these twoangles from 180°, we get 48°24´ for angle ACB.Next, solve for sides BC and AC by using the lawof sines as follows:Now that you know how to find the length of BC,you can proceed in the same manner to determine thelengths of BD and CD. Knowing the length of CD, youcan proceed in the same manner to determine the lengthsof CE and DE, knowing the length of DE, you candetermine the lengths of DF and EF, and so on. Youshould use this method only when locating inaccessiblepoints, not when a side of the triangle is to be used toextend control.In comparison with the other systems about to bedescribed, the chain of single triangles has twodisadvantages. In the first place, it can be used to coveronly a relatively narrow area. In the second place, itprovides no means for cross-checking computeddistances using computations made by a different route.In figure 15-15, for example, the only way to computethe length of BC is by solving the triangle ABC, the onlyway to compute the length of CD is by solving thetriangle BCD (using the length of BC previouslycomputed); and so on. In the systems about to bedescribed, a distance maybe computed by solving morethan one series of triangles.Chain of PolygonsTechnically speaking, of course, a triangle is apolygon; and therefore a chain of single triangles couldbe called a chain of polygons. However, in reference totriangulation figures, the term chain of polygonsrefersto a system in which a number of adjacent triangles arecombined to forma polygon, as shown in figure 15-16.Within each polygon the common vertex of the trianglesthat compose it is an observed triangulation station(which is not the case in the chain of quadrilateralsdescribed later).You can see how the length of any line shown canbe computed by two different routes. Assume that AB isthe base line, and you wish to determine the length ofline EF. You can compute this length by solvingFigure 15-16.—Chain of polygons.15-25