Quadrilateral Interior Angle Calculation Using Bearings

This solution determines the interior angle D of a closed quadrilateral ABCD based on the given forebearings (azimuths) of its sides, assuming measurements are clockwise from North.

Given Bearings

• Bearing of AB: $Br(AB) = 60^\circ$
• Bearing of BC: $Br(BC) = 149.87^\circ$
• Bearing of CD: $Br(CD) = 269.5^\circ$
• Bearing of DA: $Br(DA) = 20.17^\circ$

Calculating Interior Angle D

The interior angle at vertex D is found by analyzing the change in direction between the side CD (which arrives at D) and the side DA (which leaves D).

1. Find the Clockwise Turn Angle at D:

   This is the angle measured clockwise from the direction of segment CD to the direction of segment DA.

   Calculation: $ \text{Turn Angle} = Br(DA) - Br(CD) $

   $ \text{Turn Angle} = 20.17^\circ - 269.5^\circ = -249.33^\circ $

   Convert this to a positive clockwise angle (between 0° and 360°):

   $ \text{Clockwise Turn Angle} = -249.33^\circ + 360^\circ = 110.67^\circ $

2. Determine the Interior Angle at D:

   The interior angle ($\angle D$) of the quadrilateral is supplementary to the clockwise turn angle calculated.

   Calculation: $ \angle D = 180^\circ - \text{Clockwise Turn Angle} $

   $ \angle D = 180^\circ - 110.67^\circ = 69.33^\circ $

The calculated interior angle D is 69.33°. This value is in decimal degrees, rounded to two decimal places as required.