Let the length of the shorter diagonal be x:



This is a case of side-angle-side given:
Use the law of cosines on the triangle with sides 8,12, and x

x² = 8² + 12² - 2*8*12*cos(65°)
x² = 64 + 144 - 192*cos(65°)
x² = 208 - 192(.4226182617)
x² = 126.8572937
x = 11.26309432

Next we use the fact that two adjacent angles of a parallelogram
are supplementary to find that the angle on the lower right is
180° - 65° or 115°. We draw the longer diagonal, and label its
length y:



This is also a case of side-angle-side given:
Use the law of cosines on the triangle with sides 8,12, and y

y² = 8² + 12² - 2*8*12*cos(115°)
y² = 64 + 144 - 192*cos(115°)
y² = 208 - 192(-.4226182617)
y² = 208 + 192(.4226182617)

y² = 289.1427063
y = 17.000419672

Edwin