Question 414551
<pre><font face = "consolas" color = "indigo" size = 4><b>
Let the length of the shorter diagonal be x:

{{{drawing(400,3600/19,-1,18,-1,8,

line(0,0,12,0),line(0,0,8cos(65*pi/180),8sin(65*pi/180)),
green(line(8cos(65*pi/180),8sin(65*pi/180),12,0)),

line(8cos(65*pi/180),8sin(65*pi/180),12+8cos(65*pi/180),8sin(65*pi/180)),
locate(1,1.2,"65°"), arc(0,0,6,-6,0,65), locate(6,0,12), locate(1.1,3.9,8),
line(12,0,12+8cos(65*pi/180),8sin(65*pi/180)), locate(7.8,4.4,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:

{{{drawing(400,3600/19,-1,18,-1,8,

line(0,0,12,0),line(0,0,8cos(65*pi/180),8sin(65*pi/180)),
green(line(12+8cos(65*pi/180),8sin(65*pi/180),0,0)),

line(8cos(65*pi/180),8sin(65*pi/180),12+8cos(65*pi/180),8sin(65*pi/180)),
locate(10.5,1.3,"115°"), arc(12,0,5,-5,65,180), locate(6,0,12), locate(14,3.9,8),
line(12,0,12+8cos(65*pi/180),8sin(65*pi/180)), locate(7.5,4.5,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</pre>