To find the angle $\angle BEF$ such that triangle $\triangle DEG$ is isosceles with $DG = GE$, we first need to analyze the geometry of the given configuration.

Step 1: Understanding the Triangle

Given that $\triangle ABC$ is isosceles with $AB = AC$ and $\angle C = 30^\circ$:

• This implies $\angle A = \angle B = 75^\circ$ (since $180^\circ - 30^\circ = 150^\circ$ and divided equally).

Step 2: Identify Points and Relationships

1. Midpoint $M$ of $BC$:

   • This divides $BC$ into two equal segments.

2. Line Segment $BF$ parallel to $AM$:

   • Since $F$ is defined to be parallel to $AM$ extended, it implies $F$ will maintain the direction of $AM$.

3. Point $D$ on ray $AM$:

   • $FD \parallel AB$ implies the angle $\angle AFD = 180^\circ - \angle ABC = 105^\circ$.

Step 3: Establish Angles in the Configuration

To maintain the isosceles property of $\triangle DEG$:

• We need $\angle DGE = \angle EGD$.

From our previous construction:

• $\angle BEC = \angle BFA + \angle FAD$.

Since $F$ lies on line $AM$ extended and $D$ is also on that line, we can assert that the angles around $G$ must balance such that $\angle DGE = \angle BEF$.

Step 4: Finding $\angle BEF$

To ensure $\triangle DEG$ is isosceles:

1. We set $\angle DGE = x$.
2. Then $\angle DEG = x$ also.
3. We have: $\angle BEF + \angle DGE + \angle DEG = 180^\circ$ $\angle BEF + 2x = 180^\circ$ Solving gives: $\angle BEF = 180^\circ - 2x$

Final Calculation

If we set $x = 75^\circ$ (a reasonable assumption based on isosceles properties), we find: $\angle BEF = 180^\circ - 2 \times 75^\circ = 30^\circ.$

Conclusion

Thus, the value of $\angle BEF$ such that $\triangle DEG$ is isosceles with $DG = GE$ is:

$\angle BEF = 30^\circ.$

Questions for Further Exploration

1. How would changing the angle $\angle C$ affect the value of $\angle BEF$?
2. Can you derive the coordinates of points $B, C, D, E, F, G$ based on the given angles?
3. What conditions must hold for $\triangle ABC$ to remain isosceles when altered?
4. How would the properties of the triangle change if $\angle C$ were $60^\circ$ instead?
5. What is the relationship between the lengths of segments $DG$ and $GE$?

Tip

When solving geometric problems involving angles and parallel lines, sketching the figure can greatly aid in visualizing relationships and deriving necessary equations.