I'm assuming that these five angles are from a pentagon.
We know that the sum of the interior angles in a polygon is #(n-2)*180^@# where #n# is the number of sides in the polygon. Therefore, in a pentagon, the sum of the interior angles is
#(5-2)*180^@=3*180^@=540^@#
Since the angles' ratio is #5x:6x:7x:8x:10x#, we know that their sum must be #540^@#:
#5x+6x+7x+8x+10x=540^@#
#36x=540^@#
#x=15^@#
Now we can find all the angle measures by multiplying #12^@# by #5#, #6#, #7#, #8#, and #10#:
#mangle1=15^@*5=75^@#
#mangle2=15^@*6=90^@#
#mangle3=15^@*7=105^@#
#mangle4=15^@*8=120^@#
#mangle5=15^@*10=150^@#
We can check our answer by seeing if the sum of these angles is still #540^@#:
#color(red)(75^@+90^@)+color(blue)(105^@+120^@+150^@)=540^@#
#color(red)(165^@)+color(blue)(375^@)=540^@#
#color(purple)(540^@)=540^@#
Since this expression is true, our answer is correct.