(g) g is an acute angle of a right triangle with the leg adjacent to g 10 and the hypotenuse 20.
Since the leg is 1/2 of the hypotenuse, the triangle is a 30-60-90 right triangle. The angle opposite the shorter leg (1/2 the hypotenuse) is `30^@` while the angle opposite the longer leg is `60^@` .
(h) In the same triangle, h is the longer leg. The sides of the triangle are in the proportion `1:sqrt(3):2` . So h is `sqrt(3)` times the short leg.
(i) We have a 45-45-90 right triangle (isosceles) with hypotenuse `sqrt(6)` and i is the length of a leg.
You can use the Pythagorean theorem `i^2+i^2=(sqrt(6))^2 ==> 2i^2=6==>i^2=3 ==> i=sqrt(3)`
or you can use the shortcut that the length of a leg is the length of the hypotenuse divided by `sqrt(2)` : `i=sqrt(6)/sqrt(2)=sqrt(3)` .
(j) We have a 30-60-90 right triangle with `i=sqrt(3)` opposite the 30 degree angle and j opposite the 60 degree angle.
The length of the longer leg is the length of the shorter leg times `sqrt(3)` . Thus `j=sqrt(3)sqrt(3)=3`