Question

In trapezoid NDKF, NL and DG are altitudes, m∠F=45º, m∠K=30º, DG=8, and ND=10. Find DK, GK, FN, m∠GDK, m∠FNL, and the perimeter

Answer 1

GK - solve using elementary trig (SOHCAHTOA)
DK - Solve using pythagoras with your newly found GK
FN - solve using pythagoras
Angle GDK can be solved by the assumption that all angles inside a triangle add up to 180 degrees. You can subtract 90 (right angle) and the 30 degrees to get your answer.
Angle FNL can be found similarly.

The perimeter should be easy to calculate now.

Answer 2

GD/DK = sin30
8/DK = 1/2 (or) DK = 16

GD/GK = tan30
8/GK = 1/srqt3
GK = 8sqrt3

LN/FN = sin45
8/FN = 1/sqrt2
FN = 8sqrt2

In triangle GDK,

90 + 30 + angleGDK = 180

Angle GDK = 60

In traingle FNL,

90 + 45 + angleFNL = 180

angleFNL = 45

Perimeter = ND + DK + GK + GL + FL + FN = 10 + 16 + 8sqrt3 + 10 + 8 + 8sqrt2 = 34 + sqrt3 + sqrt2