To solve the problem, we need to use the properties of isosceles triangles and the sum of angles in a triangle.

Step-by-step solution:

1. The triangle $\triangle DEF$ is isosceles since the two marked sides are equal. Therefore, the base angles $\angle D$ and $\angle F$ are equal. Let these angles each be $x^\circ$.

2. The sum of angles in any triangle is $180^\circ$. Thus: $\angle D + \angle E + \angle F = 180^\circ$ Substituting $\angle D = x^\circ$, $\angle F = x^\circ$, and $\angle E = 30^\circ$, we get: $x + x + 30 = 180$

3. Simplify the equation: $2x + 30 = 180$

4. Solve for $x$: $2x = 150 \implies x = 75$

Final Answer:

The value of $x$ is C. 75.

Would you like further clarification or details on any of the steps?

Related Questions:

1. How can the properties of isosceles triangles help solve similar problems?
2. What is the sum of angles in any polygon, and how is it calculated?
3. Can you generalize the relationship between base angles in isosceles triangles?
4. How would the solution change if $\angle E$ were $60^\circ$ instead of $30^\circ$?
5. What are the characteristics of equilateral triangles compared to isosceles triangles?

Tip:

Always verify if a triangle is isosceles or equilateral when you see marked equal sides, as it simplifies calculations significantly.