Question

The forebearing of four sides of a closed quadrilateral ABCDA are
AB = 60°, BC = 149.87°, CD = 269.5°, DA = 20.17°.
The calculated value of interior angle D is _____° (Answer in decimal degrees and rounded off to two decimal values).

Hint:

Use back bearing of the incoming line minus fore bearing of the outgoing line at D, then check that all four interior angles of the quadrilateral sum to 360 degrees.

Answer 1

Correct Answer: 69.31

Step 1: Recall how interior angles are obtained from consecutive bearings in a traverse.
In a closed traverse ABCDA run in the order A to B to C to D to A, the interior angle at any station is found from the fore bearing (FB) of the line leaving that station and the back bearing (BB) of the line arriving at that station, using \( \text{Interior angle} = BB(\text{incoming line}) - FB(\text{outgoing line}) \), where the back bearing of a line is obtained by adding or subtracting \(180^\circ\) from its fore bearing so that the result stays within \(0^\circ\) to \(360^\circ\): \( BB = FB \pm 180^\circ \).
Step 2: Identify the lines meeting at D.
At station D, the line arriving is CD (fore bearing \(269.5^\circ\)) and the line leaving is DA (fore bearing \(20.17^\circ\)).
Step 3: Compute the back bearing of the incoming line CD.
Since \(FB(CD) = 269.5^\circ > 180^\circ\), \( BB(CD) = FB(CD) - 180^\circ = 269.5^\circ - 180^\circ = 89.5^\circ \).
Step 4: Compute the interior angle at D.
\( \angle D = BB(CD) - FB(DA) = 89.5^\circ - 20.17^\circ = 69.33^\circ \).
Step 5: Verify using the angle sum check for a quadrilateral.
Applying the same method at the other three stations, \( \angle B = BB(AB) - FB(BC) = (60^\circ+180^\circ) - 149.87^\circ = 240^\circ - 149.87^\circ = 90.13^\circ \), \( \angle C = BB(BC) - FB(CD) = (149.87^\circ+180^\circ) - 269.5^\circ = 329.87^\circ - 269.5^\circ = 60.37^\circ \), and \( \angle A = BB(DA) - FB(AB) = (20.17^\circ+180^\circ) - 60^\circ = 200.17^\circ - 60^\circ = 140.17^\circ \). The sum \(90.13^\circ + 60.37^\circ + 69.33^\circ + 140.17^\circ = 360.00^\circ\), which correctly equals the theoretical sum of interior angles of a quadrilateral, \((n-2)\times180^\circ = 360^\circ\), confirming the computation of \(\angle D\) is consistent.
\[ \boxed{\angle D = 69.33^\circ \; (\text{accepted range } 69.31^\circ \text{ to } 69.35^\circ)} \]