Diagonals of a parallelogram bisect each other.
They divide a parallelogram in 4 (four) small triangles.
The angles between the diagonals are vertical and supplementary.
So, if    is an acute angle between the diagonals, 
then the other, supplementary angle    is  .


Let's find an acute angle    between the diagonals for the given parallelogram.
Obviously, this angle is opposite to the shorter side of the parallelogram.
The halves of diagonals are 16/2 = 8 cm  and  24/2 = 12 cm long, and acute angle    
is a concluded angle between the halves of diagonals.


Write the cosine law for such a triangle

    10^2 =  +  - .


It gives

     =  =  = .


These four small triangles of the subdivision have a remarkable property:
their areas are all the same.


It is easy to prove:  the area of each such a triangle is half the product of the sides
that are halves of diagonals, by the sine of the angle between them.

    area =   for the acute angle between diagonals

and  

    area =   for the obtuse angle between diagonals.


But the acute angle and the obtuse angle are SUPPLEMENTARY, so, they have the same value of sine.


GREAT !


Let's calculate the sine of an angle  between the diagonals.


For the acute angle, it is  

     =  =  =  =  = .


For the obtuse angle,  the sine value is the same,  .


So, each of the four small triangle of the subdivision has the area

     =  = .


The area of the whole parallelogram is then four times the area of each separate small triangle 
of the subdivision.

So, the area of the parallelogram is   = ,  or about  158.7450787 cm^2.



To find the measure of the longer side, apply the cosine law again.
Use the obtuse-angled triangle of the subdivision.
Its sides are 8 cm and 12 cm;  the cosine of the obtuse concluded angle  = 
is  = .


So, we write for the square of the longer side of the parallelogram

      = 64+144+108 = 316.


Thus the longer side of the parallelogram is   = 17.77638883 cm (approximately).



The last question is to find the smaller angle of the parallelogram,  .

Let's find it, using the shorter side of 10 cm and what we just learned about this parallelogram: 
its area is    and its longer side is  .


The area of any parallelogram is the product of any two adjacent sides by the sine of the concluded angle.


So, for   we can write this equation

    } = .


Thus   =  =  = 0.893010837.


Hence,   = arcsin(0.893010837) = 63.2540602 degrees.


ANSWER.  The area of the parallelogram is  ,  or about  158.7450787 cm^2.

         The measure of the longer side is  ,  or  17.77638883 cm (approximately).

         The measure of the smaller angle of the parallelogram is  arcsin(0.893010837),  or about 63.2540602 degrees.