To solve for the value of $x$, we will use the fact that the sum of the angles around point $E$ equals 360°.

Step 1: Identify the known angles

The angles around point $E$ include:

• $\angle AEG = x^\circ$ (unknown)
• $\angle FED = 80^\circ$
• $\angle DEB = 20^\circ$

Let the remaining angle, $\angle CEF$, be denoted by $y^\circ$.

Step 2: Use the angle sum around point $E$

The sum of the angles around point $E$ is: $x^\circ + 80^\circ + 20^\circ + y^\circ = 360^\circ$

Step 3: Simplify the equation

We can simplify the equation as follows: $x + 100^\circ + y = 360^\circ$ Subtracting 100° from both sides: $x + y = 260^\circ$

Step 4: Use the linear pair at $\angle FED$

$\angle FED$ and $\angle CEF$ are a linear pair and must add up to 180°: $80^\circ + y = 180^\circ$ Thus, $y = 100^\circ$.

Step 5: Substitute $y$ back into the sum equation

Substituting $y = 100^\circ$ into the equation $x + y = 260^\circ$: $x + 100^\circ = 260^\circ$ Subtracting 100° from both sides: $x = 160^\circ$

Final Answer:

$x = 160^\circ$

Would you like more details, or do you have any questions?

---

Here are 8 related questions you might find helpful:

1. How do you calculate the sum of angles around a point?
2. What are supplementary angles, and how are they used in problems like this?
3. What is a linear pair, and why do they sum to 180 degrees?
4. How can the angles in a triangle help solve related problems?
5. What are vertical angles, and how do they relate to intersection points?
6. How does one identify angles on parallel lines cut by a transversal?
7. Can angle bisectors be helpful in problems involving intersecting lines?
8. How does the sum of angles in a quadrilateral compare to this problem?

Tip: Always check if angles form a complete circle or a straight line as a quick way to simplify problems!