We are given a five-sided polygon $ABCDE$ and need to find the measure of angle $x$ in the figure.

Step-by-step Solution:

1. Understanding the diagram:

   • We know the internal angles at points $D$, $C$, and $B$:
     • $\angle D = 115^\circ$
     • $\angle C = 80^\circ$
     • $\angle B = 65^\circ$
   • The angle at $E$ is unknown, and the exterior angle at point $A$ is adjacent to $\angle B = 65^\circ$, implying the straight angle formed is $180^\circ$.

2. Sum of interior angles of the pentagon: The sum of the interior angles of any polygon is given by the formula: $S = (n - 2) \times 180^\circ$ where $n$ is the number of sides. For a pentagon ($n = 5$): $S = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$

3. Setting up the equation: We know four of the five angles (except $x$). The sum of the known angles is: $115^\circ + 80^\circ + 65^\circ = 260^\circ$ Therefore, the remaining two angles (i.e., $\angle x$ and $\angle E$) must add up to: $540^\circ - 260^\circ = 280^\circ$

4. Using external angles: Since $\angle A$ is part of a straight line, the exterior angle at $A$ is: $\text{Exterior Angle at A} = 180^\circ - 65^\circ = 115^\circ$ This is angle $E$.

5. Finding $x$: Finally, the remaining angle $x$ is: $x = 280^\circ - 115^\circ = 165^\circ$

However, based on the choices given (a. 60°, b. 80°, c. 120°, d. 140°), the correct answer appears to be c. 120°, which should be checked in context.

Would you like further details or clarifications?

Here are 5 related questions you might want to explore:

1. How do you calculate the sum of interior angles for different polygons?
2. What is the relationship between interior and exterior angles in polygons?
3. How does the exterior angle theorem apply to pentagons?
4. How do different types of pentagons affect angle properties (e.g., regular vs irregular)?
5. What techniques are useful for solving geometric problems involving multiple angles?

Tip: Always double-check the sum of angles in polygons to ensure your solution is consistent with known geometric properties.