A quadrilateral, too, is technically a polygon; and a chain of quadrilaterals would be technically a chain of polygons. However, with reference to triangulation figures, the term chain of quadrilaterals refers to a figure arrangement like that shown in figure 15-17. Within each of the quadrilaterals shown, the triangles on which computations are based are not the four adjacent triangles visible to the eye, but four overlapping triangleseach of which has as sides two sides of the quadrilateral and one diagonal of the quadrilateral. For example, in quadrilateral ACDB there are four overlapping triangles as follows: ADC, ADB, ABC, and BCD. You can see that solving these four triangles will give you two computations for the length of each unknown side of the quadrilateral.

Consider, for example, the quadrilateral ACDB. Look at angle BAC. We will call the whole angle at a comer by the letter (as, angle A) and a less-than-whole angle at a corner by the number shown (as, angle 1). The angles at each station on the quadrilateral, as measured with a protractor to the nearest 0.5 degree and estimated to the nearest 0.1 degree, are sized as follows:

To solve the quadrilateral, you solve each of the overlapping triangles. First, you solve triangle ABC for AC and BC, using the law of sines as follows:

Then, using similar computation procedures, you solve triangle ABD for sides BD and AD, triangle ADC for AC and CD, and triangle BCD for BD and CD. The solutions for each of the overlapping triangles are summarized as follows:

As you can see, for each of the unknown sides of the quadrilateral (AC, CD, and BD), values have been obtained by two different routes. You can also see that there are discrepancies in the values, almost the same for AC and BD and smaller for CD. All the discrepancies shown are much larger than would be tolerable in actual practice; they reflect the high imprecision of the original protractor measurement of the angles. The example has been given here only to illustrate the basic principles and procedures of chain-of-quadrilateral triangulation.