Quadrilaterals are four sided polygons, with four vertexes, whose total interior angles add up to 360 degrees. The most common quadrilaterals are the rectangle, square, trapezoid, rhombus, and parallelogram. Finding the interior angles of a quadrilateral is a relatively simple process, and can be done if three angles, two angles, or one angle and four sides are known. By dividing a quadrilateral into two triangles, any unknown angle can be found if one of the three conditions are true.
Divide the quadrilateral into two triangles. You will need to split two of the angles in half when you divide the quadrilateral. For example if you had an angle of 60 degrees it will become 30 degrees on both sides of the dividing line.
Add the sum of the angles for the triangle with the missing angle. For example if one of the quadrilateral's triangles had the angles 30 and 50 degrees, you would add them together to get 80 degrees (30 + 50 = 80).
Subtract the sum of the angles from 180 degrees to get the missing angle. For example if a triangle in a quadrilateral had the angles of 30 and 50 degrees, you would have a third angle equal to 100 degrees (180 - 80 = 100).
Divide the quadrilateral in half to form two triangles. Always try to divide the quadrilateral in half by splitting one of the angles in half. For example, a quadrilateral with two angles of 45 degrees next to each other, you would start the dividing line from one of the 45 degree angles. If you cannot divide the quadrilateral from one of the angles, and get both angles on opposite sides of the quadrilateral, you will need to know the length of the sides of the quadrilateral, and have to use the 1 angle four sides known process.
Add the the sum of the angles in the triangle with two angles. For example, if you have a triangle inside a quadrilateral with the angles 45 and 20 degrees, you would get a sum of 65 degrees (20 + 45 = 65).
Subtract the sum of the angles from 180 to get the third angle of the triangle. For example, if you have a triangle within a quadrilateral that has the angles 20 and 45 degrees you would get a third angle of 115 degrees (180 - 65 = 115).
Add the two known angles of the quadrilateral with the new angle. For example if your quadrilateral had the angles 45, 40, and 115 degrees, you would get a sum of 200 degrees (45 + 40 + 115 = 200).
Subtract the sum of the three angles from 360, to get the final angle. For example, a quadrilateral with the angles 40, 45, and 115 degrees, you would get a fourth angle of 160 degrees (360 - 200 = 160).
1 Angle and 4 Sides
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).