To find the value of $x$ in the diagram, let's go through the information given and use the properties of congruent triangles.

1. Given Information:

   • $\triangle ABC \cong \triangle DEF$, which means all corresponding angles and sides of these triangles are equal.
   • In $\triangle ABC$, we know that $\angle A = 65^\circ$ and $\angle C = 55^\circ$.
   • Since the triangles are congruent, the angles in $\triangle DEF$ will match those in $\triangle ABC$.

2. Finding the Unknown Angle in $\triangle ABC$: Since the sum of angles in a triangle is $180^\circ$, we can find $\angle B$ as follows: $\angle B = 180^\circ - (65^\circ + 55^\circ) = 180^\circ - 120^\circ = 60^\circ$

3. Corresponding Angles in $\triangle DEF$:

   • Since $\triangle ABC \cong \triangle DEF$, $\angle D$ (corresponding to $\angle A$) is $65^\circ$, $\angle F$ (corresponding to $\angle C$) is $55^\circ$, and $\angle E$ (corresponding to $\angle B$) is $60^\circ$.

4. Setting Up the Equation for $x$:

   • According to the diagram, $\angle E$ is marked as $5x$.
   • Since $\angle E = 60^\circ$, we have: $5x = 60^\circ$

5. Solving for $x$: $x = \frac{60^\circ}{5} = 12^\circ$

Final Answer:

The value of $x$ is $12$.

Would you like further details on this solution or have any questions?

Here are some related questions that might deepen your understanding:

1. How can we confirm that $\triangle ABC \cong \triangle DEF$ based on the given angles?
2. What are other methods to verify congruency between two triangles?
3. How does the concept of congruent triangles apply to real-world problems?
4. What if the value of $5x$ was different; how would that affect the congruency?
5. Can congruence be determined by only comparing two angles in each triangle?

Tip: Always remember that in congruent triangles, corresponding angles are equal and corresponding sides are proportional, making them a useful tool for solving geometric problems.