Question

I need help on this question

Answer 1

`\angle\,DGF=30^o`

which agrees with the first answer listed among the possible options.

Explanation:

Notice that if a circle is perfectly inscribed inside the triangle, the triangle must have the symmetries that correspond to an equilateral triangle (all equal sides). Such type of triangles have the three angles equal as well, and since they need to add to
`180^o`, they must measured each
`60^o`

Now, the segment that joins the center of this symmetric triangle (also the center of the circle) with the triangle's vertices, has to bisect each each internal angle in equal parts.

Therefore, angle
`\angle\,DGF` must measure half of
`60^o`, that is:
`30^o`