Hi gobbajeezalus

(Assuming this is a right angled triangle) all unknown angles are available to us from basic properties of triangles so

the unknown angle on the base is 180 - 38 =142

from that we know the unknown angle in this triangle above is

180 - 142 - 27 =11

The larger angle for the border triangle is 63 leaving the last unknown angle in the top corner as 52

From there you just use the sine rule to find the hypotenuse of the larger triangle and you can then use pythagoras to find the value for h, basically

sine rule

hyp/sin 142 = 500/sin 11

hypotenuse = 1613feet

pythagoras

sin 27 =h/1613

h = 732.4 feet

Hope thats right