form vertically opposite angles 9-10 and 11-12. Fromyour knowledge of geometry, you know that when twostraight lines intersect the vertically opposite anglesthus formed are equal. From the fact that the sum of theangles in any triangle is 180°, it follows that for any pairof vertically opposite angles in figure 15-28, the sumsof the other two angles in each of the correspondingtriangles must be equal.For example: In figure 15-28, angles 11 and 12 areequal vertically opposite angles. Angle 11 lies in atriangle in which the other two angles are angles 1 and8; angle 12 lies in a triangle in which the other two anglesare angles 4 and 5. It follows, then, that the sum of angle1 plus angle 8 must equal the sum of angle 5 plus angle4. By similar reasoning, the sum of angle 2 plus angle 3must equal the sum of angle 6 plus angle 7.Suppose now, that the values of angles 2, 3, 6, and7, after adjustment for the sum of interior angles, areasfollows:The difference between the two sums is 8 seconds.This means that, to make the sums equal, 4 secondsshould be subtracted from the 2-3 sum and added to the6-7 sum. To subtract 4 seconds from the 2-3 sum, yousubtract 2 seconds from each angle; to add 4 seconds tothe 6-7 sum, you add 2 seconds to each angle.The final step in quadrilateral adjustment is relatedto the fact that you can compute the length of a side ina quadrilateral by more than one route. The final step inadjustment is to ensure that, for a given side, you willget the same result, to the desired number of significantfigures, regardless of the route your computations take.This final adjustment is called the log-sineadjustment, because it uses the logarithmic sines of theangles. The method is based on the use of side equationsto derive an equation from which the sides areeliminated and only the sines of the angles remain. Thisequation is derived as follows:Suppose that in figure 15-28,AB is the baseline andthe length of CD is to be computed. By the law of sines,By the same law,Substituting the value of AD, we haveAgain by the law of sines we haveBy the same law,Substituting this value for BC, we haveWe now have two values for CD, as follows:It follows thatCanceling out AB, we haveBy the law of proportions, this can be expressed as15-36