In the given figure, $$\angle A \colon \angle B \colon \angle C=14 \colon 3 \colon 1$$. The two additional lines are drawn as described in the figure. The measure of $$\angle EGF$$ in degrees is
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The angle ratio gives $$\angle A=140^\circ$$, $$\angle B=30^\circ$$ and $$\angle C=10^\circ$$. The exterior angle $$\angle CAD$$ is $$40^\circ$$, so the line through $$A$$ makes $$10^\circ$$ with $$AC$$ and is parallel to $$BC$$. The line through $$B$$ also makes $$10^\circ$$ with $$BC$$, so the vertically opposite angle $$\angle EGF$$ is $$10^\circ$$.