Solve for g, h, i and j.

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(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^@` .

`g=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.

`h=10sqrt(3)`

(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)` .

`i=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`

`j=3`

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