How to find angles of quadrilateral if sides and one angle are given?
Answers
Divide the quadrilateral in half to form two triangles. It is a good idea to divide it in half at the known angle to give you an angle to work with in both triangles. For example if you had a quadrilateral with a known angle of 40 degrees, by dividing the angle in half you have 20 degrees to work with on both sides.
Divide the sine of the known angle in both triangles by the length of the opposing side. For example if you have a two triangles with a angle of 20 degrees and an opposing side of 10 inside a quadrilateral, you would get a quotient of 0.03 (sin20 / 10 = 0.03).
Multiply the quotient of the sine of the known angle divided by it's opposing side by the other known side of the triangle. Do this for both triangles. For example, two triangles inside of a quadrilateral with known angles of 20 and opposing sides of 10 and another side of 5, would have a product of 0.15 for both triangles (0.03 x 5 = 0.15).
Find the cosecant of the product for both triangles, this number will be the length of the dividing line that forms the hypotenuse. The cosecant is often found on calculators as either "csc", "asin", or "sin^-1". For example the cosecant of 0.15 would be 8.63 (csc15 = 8.63).
Add the squares for the two sides forming and unknown angle, and subtract them by the square of the opposing side of the unknown angle. For example if two triangles in a quadrilateral, had an two sides of 5 and 10 creating an opposing angle to a side equal to 8.63, you would get a difference of 50.52 ((10 x 10) + (5 x 5) - (8.63 - 8.63) = 50.52)
Divide the difference by the product of the two sides that form the unknown angle and 2. For example, two triangles inside a quadrilateral with two sides of 5 and 10 that form an unknown angle with an opposing side of 8.63, would have a quotient of 0.51 (50.52 / (10 x 5 x 2) = 0.51).
Find the secant of the quotient to find the unknown angle. For example the secant of 0.51 would create an angle of 59.34 degrees.
Add the sum of all three angles in the quadrilateral and subtract it from 360 to get the final angle. For example a quadrilateral with the angles 40, 59.34, and 59.34 degrees would have a fourth angle of 201.32 degrees (360 - (59.34 + 59.34 + 40) = 201.32).
Answer:
William has answered the easy part. The hard part is finding the shape of the quadrilateral that has the area you have been given. Because a quadrilateral can be squashed in various directions. unlike a triangle, two quadrilaterals with corresponding sides of the same length can have different areas. I don't happen to know how to get the correct quadrilateral, but it is one of those things that must have been solved, perhaps even by the Greeks.