We have a parallelogram with sides of 8cm and 5cm and an angle of 140 degrees, calculate the length of two diagonals

We would first like to work out the size of all the angles which have not been give, as one angle is 140 degrees we know the angle opposite to it is also 140 degrees due to the symetry of the shape. The two angles add to 280 degrees, this means that as the interior angles of a square add to 360 degrees, this leaves us with the other two angles adding to 80 degrees thus by utilising the same theory, we can say that the other two angles are both 40 degrees. Now we can apply the cosine rule, a^2 = b^2 + c^2 - 2bcCosA, for the first triangle, using the angle of 140 degrees, b is 5, c is 8  and A is 140 degrees (the b and the c are interchangeable however the missing length is always opposite to the known angle), now we can substitute the number into the formula and we have an a^2 value of 150.3 (3sf) and so the diagonal a, is 12.3cm. We can apply the same theory for the other diagonal to get a length of 5.3cm (3sf)