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let's work an angular closure problem in this scenario we have a five-sided traverse connecting in a closed polygon points ABCD and E and the direction of side a B is known and the angles that you see here that are interior to the polygon are measured angles therefore as the scope at the left says we're going to adjust the interior angles of this traverse adjust the errors and we're going to compute the azimuth of all the Traverse sides expressed in the counterclockwise direction we want to compute this in the counterclockwise direction counter clockwise of course runs this way doesn't it so we want to be consistent with this pattern that's established here for the direction of a be that direction goes this way thus north as we see it is at the top of the drawing here okay so if we're going counterclockwise that means we need to come up with the direction going this way this way this way and this way for the remaining four sides before we can do that we need to adjust the angles so first we need to determine the angular closure we have a fixed relationship of the number of angles in or I should say the number of sides which would be the same for a closed polygon the number of angles in a closed polygon -2 times 180 degrees will predict the nut the number of degrees of the sum of the interior angles in a perfect polygon what do I mean by that let's say it this way the sum of those interior make a little angle symbol the sum of the interior angles equals n minus 2 over or excuse me times 180 180 degrees and N equals the number of sides or the number of angles so in this case we have five sides so if I plug N equals five in here for N equals 5 then the sum of the interior angles would be n minus 2 that's three times 180 that is going to give us 540 degrees okay so we need to find out how close to that we actually got there is error and every measurement therefore we're going to find the sum of the errors the the sum of the effects of the errors I may cancel with each other well the way we do that is simply take every last one of these and sum them up and in a perfect world they would add to 540 degrees but we know they're not going to so rather than write them out and spell it out for you I encourage you to take a moment or two maybe pause this video and add those up right now and see what you get the sum that I get is 539 degrees 59 minutes and 39 seconds okay so if I subtract that from or subtract 540 degrees from that I will get 0 degrees 0 minutes and 21 seconds that's a negative angular error in this case because it is less than 540 that's the sum of the angular errors in these five angles well we correct that the simplest way to correct that simplest accepted way is to distribute that error evenly among all the angles so I'm gonna take my 21 seconds I'm gonna take my 21 seconds here and divide that by my 5 angles okay so that gives me that gives me a correction of 4.2 now consider I almost forgot if we have a negative 21 second error I'm going to take the opposite in sign of that to get my correction so here I would have a positive 4.2 second correction per angle now in reality dividing down to the tenth of a second may be overkill so it's not uncommon that we could say I've got five angles I'm going to apply a four second correction on four of those and a five-second correction on the last one this is simply an acknowledgement that on small traverses this may it may be more trouble than it's worth if you're doing anything by by hand to go with tenth of a second for our purposes today I'm going to do that so these are the corrections i will apply apply to each angle so i think i'm going to start here at angle B and I'm going to apply four seconds to each angle remember I am overall I'm this many seconds to few so I'm going to have to add seconds to each angle to get the sum back up to 540 so I'm going to add four seconds to this one I'm going to add four seconds for this one four seconds here four seconds here and the last one I will add five seconds to you'll find that on very small traverse is like most of us run I'm talking about a traverse that's a you know a mile across by a mile across the seconds don't add up to much at all so these are my corrected corrected interior angles so now I need to calculate interior angles excuse me azimuth in order to calculate azimuth I have to start with something that I know and I know the direction from A to B it is 94 degrees 16 minutes 27 seconds so if I'm going to proceed counterclockwise the next course whose direction I will calculate will be from B to C and you can see it here B to C that will be the next one and the next one will be C to D and so forth so in order to make this happen in azimuth computations I can take the back azimuth of line a B which is the course previous to the one I'm going to do next and then I'm going to add the interior angle ABC B being the vertex a being the back side as we'd call it and C being the foresight so B is the vertex of that angle when I add those I will get the azimuth of line BC and then I'm going to do this over and over and over again so let's start with some math here so my starting azimuth was 94 16:27 that is the azimuth of a B well I need to either add or subtract 180 to get the back azimuth of that so when I add 180 to this I will get to 74 16:27 and that is the back azimuth of a B okay that is the direction going now from B to a isn't it the back azimuth of a B is the same as the azimuth of BA and in this case I'm going to add the interior angle now this is the adjusted interior angle so I have to make sure I use the adjusted seconds so in this case I'm using 56 51 55 okay that is angle a B C isn't it so then when I apply this carefully I will get three 3108 22 that is the azimuth of BC does that seem to make sense I think so because here if I look carefully angle the direction from B to C is going this way three three one zero 822 that's headed off in the northwest quadrant yep that seems pretty reasonable okay so I'm going to continue so now I have to come up with the back azimuth of BC so the back azimuth of BC I can get that by either adding or subtracting this time I'm going to subtract whether I add or subtract isn't really critical because you can still end up with the same result you may take more steps going one way or the other but with experience you start to see the pattern so when I come up with this difference what's that gonna be that's gonna be one five one zero 822 that's the back azimuth BC I'm going to add the interior angle at C so let's see what that looks like that interior angle at C is 233 3855 right I remember I'm using my adjusted angle so this is angle B C D isn't it well when I do this I'm gonna come up with an answer that is 384 47:17 right yep 17 but is that a valid azimuth no because a valid azimuth is between 0 and 360 degrees so I simply subtract 360 degrees and this becomes 24 47:17 so this is now the azimuth CD isn't it okay well I think we're on roll let's keep going so I need a back azimuth and in this case I am adding now the angle at D aren't I because I am between line CD and de so this is 64 57 56 this is angle C D E and that gives me 269 4513 that is now azimuth de isn't it well it won't take me long and I'll have this finished so now I need the back azimuth very repetitive now isn't it okay I think you can see that's eighty nine forty five thirteen that's the back azimuth of de and then I'm going to add 94 55 and my corrected value was 19 wasn't it and this is angle D e a so now when I do the math I will get 184 4 0 3 2 that is the azimuth ei okay now here's here's where we stand let me let me put all of this summarized up here for you we have azimuth C D we said was let's see 24 see if I can write on my side like this 47 17 right 24:47 17 and then de was whoops 2 6 9 45 13 and then a e-excuse me ei was 184 40 32 well I've got all five sides now but is there any chance I've made a computation error sure there is so in order to prove to myself that I have done it right I'm going to use the fifth interior angle you see I have used this one and this one and this one and this one but I haven't yet used this one so what I will do is calculate the direction going from A to B and see if it indeed does come out to be the same thing that I started with so let's do exactly that here I've got I've got my azimuth from E to a well I can come up with a back azimuth of that that's 180 0 0 0 0 is the value I will subtract and that gives me 440 32 and then I'm going to add this interior angle just as we have done on the others 89 35 55 correct and thus that was angle EAB and lo and behold it does come out the same 94 16 27 so that is azimuth a B yes this checks it is okay so that's that is how we adjust interior angles on the Traverse and we can have computed all the Traverse sides expressed in the counterclockwise direction just one thing I want to remind you on when we work in the counterclockwise direction we add interior angles in a process like this and when we are going clockwise instead we subtract the interior angles there is a difference here so simple rule that you have to apply consistently so before you start the problem next time you do this ask yourself which direction am I going because that will determine whether you are adding interior angles or
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