To solve the given problems, let’s analyze the information provided:

1. Incenter Definition: $G$ is the incenter of $\triangle ABC$, meaning it is the point where the angle bisectors of $\triangle ABC$ intersect. This also means that $G$ is equidistant from each side of the triangle and each of the segments $DG$, $EG$, and $FG$ represents the inradius of the triangle, marked as 4 units.

2. Understanding Given Information:

   • $\angle ABC = 20^\circ$.
   • $\angle ACB = 11^\circ$.
   • $AG = DG = FG = 4$ (as $G$ is equidistant to each side due to its position as the incenter).

3. Finding Measures:

   • m∠ABG (10): Since $G$ is the incenter, $\angle ABG$ is half of $\angle ABC$, so $m\angle ABG = \frac{20^\circ}{2} = 10^\circ$.
   • m∠BCA (11): This is given directly as $11^\circ$.
   • m∠BAC (12): To find $\angle BAC$, we use the triangle angle sum property: $\angle BAC = 180^\circ - \angle ABC - \angle BCA = 180^\circ - 20^\circ - 11^\circ = 149^\circ.$
   • m∠BAG (13): Since $G$ is the incenter and $\angle BAC$ is bisected by $AG$, $m\angle BAG = \frac{149^\circ}{2} = 74.5^\circ$.
   • DG (14): $DG$ is given as 4 units, as it's a radius of the incircle.
   • BE (15): $BE$ is also a radius of the incircle, so $BE = DG = 4$ units.
   • BG (16): There’s no further information to deduce $BG$ directly from the diagram, but typically more details about triangle side lengths would be needed.
   • GC (17): Similarly, $GC$ cannot be determined with the given information alone without additional side length or angle measures.

I hope this clarifies the approach for solving each part. Do you have questions on any specific calculation, or would you like further details?

Follow-up Questions:

1. How would you determine $BG$ or $GC$ if more information was given?
2. What properties of an incenter can help in solving triangle-related problems?
3. Why does the incenter equally divide the angles of a triangle?
4. How does knowing one angle of a triangle affect the calculations for other angles?
5. Can we find the area of $\triangle ABC$ given the information here?

Tip: Remember that the incenter is always equidistant from each side of a triangle, which can help when calculating side or segment lengths.