Lines AB, CD and FG intersect each other at point E. Angle AEG and FED are labeled as x degrees and 80 degrees respectively. Angle DEB is labeled 20 degrees.
\[A\]
\[B\]
\[C\]
\[D\]
\[E\]
\[F\]
\[G\]
\[20^\circ\]
\[80^\circ\]
\[x^\circ\]
NOTE: Angles not necessarily drawn to scale.
\[x = \]

\[\Large{{}^\circ}\]

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?

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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!

Learn how to solve a problem involving intersecting lines and angles. Includes step-by-step calculation of angles around a point and the use of linear pair concept. Suitable for high school students studying geometry.

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