Given a closed traverse with $3$ sides as shown below. It is known that the bearing angle of line $AB$ is $65{35}^{\text{'}}{56}^{\text{'}\text{'}}$ NE$.$ Based on the measurements, the observed interior angle at A is $55{56}^{\text{'}}{40}^{\text{'}\text{'}}$; the observed interior angle at Point B is $800{1220}^{\text{'}\text{'}},$ and the observed interior angle at Point $C$ is $43{51}^{\text{'}}{30}^{\text{'}\text{'}}.$

Based on the onsite measurement, the lengths of two sides: $AB$ and $AC,$ are found to be $122\mathrm{.}450ft$ and $174\mathrm{.}120ft,$ respectively according to the measurements. However, due to some reasons, the length of $BC$ cannot be measured directly. Answer the following questions.

$($a$)$ Based on the observed interior angles, what is the error of closure? $(3$ pt$)$

$($b$)$ Based on your answer of $($a$),$ balance three interior angles evenly, and give the balanced interior angles at Point A$,$ B and C$,$ respectively. Fill the following table. $(2$ pt$)$

$\backslash$table$[[$Station$,$Observed Interior Angle,Balanced Interior Angle$],[$A$,55{56}^{\text{'}}{40}^{\text{'}\text{'}},],[B,80{12}^{\text{'}}{20}^{\text{'}\text{'}},],[C,43{51}^{\text{'}}{30}^{\text{'}\text{'}},]]$

VEN$2370$

$5$