SOLUTION: Please answer this is really hard. The observed angles of a quadrilateral after station and side adjusmtments are: Angle DBA=30 Degrees angle CBD=48 Degrees angle BCA=59 Degrees an

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Question 1074041: Please answer this is really hard. The observed angles of a quadrilateral after station and side adjusmtments are: Angle DBA=30 Degrees angle CBD=48 Degrees angle BCA=59 Degrees and angle DCA =21 degrees. Compute the angles BDA,DAC and DAB
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I do not know what was meant by "station and side adjustments."
I imagine the quadrilateral (with its diagonals) looks like this,
or a rotated and/or flipped version.

DBA=30%5Eo , CBD=48%5Eo , BCA=59%5Eo , and DCA+=21%5Eo .
So, BCD=BCA%2BDCA=59%5Eo%2B21%5Eo=80%5Eo and CBA=CBD%2BDBA=48%5Eo%2B30%5Eo=78%5Eo
Also, DXA=CXB=180%5Eo-59%5Eo-48%5Eo=73%5Eo and
BAC=180%5Eo-59%5Eo-78%5Eo=43%5Eo based on the sum of angle measures for triangles BXC and BAC respectively.

With that, I can compute approximate measures for BDA, DAC and DAB.
It takes quite a bit of coordinate geometry calculations, and I do not get an exact value, but it is the best I can do so far.
For ease of calculations, I set coordinates for points C and B as
C%280%2C0%29 and B%2810%2C0%29 .
Then I find coordinates for D%28x%5BD%5D%2Cy%5BD%5D%29 and A%28x%5BA%5D%2Cy%5BA%5D%29 .
system%28y%5BD%5Dcot%2880%5Eo%29=x%5BD%5D%2Cy%5BD%5Dcot%2848%5Eo%29=10-x%5BD%5D%29 --> --> -->
Approximate value: D%281.638%2C9.287%29
Similarly for A,

Approximate value: A%287.387%2C12.294%29 .
(I have my computer keep more digits in these calculations, because they are intermediate calculations, and I round at the end).
The slope of line DA is the tangent of the angle DA makes with the x-axis (or with any horizontal line).
It is %2812.294-9.287%29%2F%287.387-1.638%29=3.006%2F5.749=0.5230 .
That is the tangent of approximately a 27.6%5Eo angle.
Since line BD makes a -48%5Eo angle with the horizontal,
the measure of angle BDA is about BDA=27.6%5Eo%2B48%5Eo=75.6%5Eo .
DAC=59%5Eo-27.6%5Eo=31.4%5Eo based on the angles DA and AC make with the x-axis.
Or, DAC=180%5Eo-73%5Eo-27.6%5Eo=31.4%5Eo based on the sum of angle measures for triangle ADX.
Finally, DAB=DAC%2BBAC=31.4%5Eo%2B43%5Eo=74.4%5Eo .