https://file.aiquickdraw.com/mathbot/file/1727211795159hvt3d6ek.jpg

The problem in the image reads:

"In the accompanying diagram, \( \overline{AB} \parallel \overline{CD} \), \( \overline{EF} \) intersects \( \overline{AB} \) at \( E \) and \( \overline{CD} \) at \( F \), and \( \overline{GH} \) intersects \( \overline{AB} \) at \( G \) and \( \overline{EF} \) at \( H \). If \( m\angle EGH = 40^\circ \), \( m\angle GHE = 80^\circ \), and \( m\angle EFD = x \), what is the value of \( x \)?"

### Solution:

We need to determine the value of \( m\angle EFD = x \).

#### Step 1: Use the information from the diagram
From the problem, we know that \( \overline{AB} \parallel \overline{CD} \) and \( \overline{EF} \) is a transversal intersecting both lines. This creates alternate interior angles.

#### Step 2: Find relationships between the given angles
- \( m\angle EGH = 40^\circ \)
- \( m\angle GHE = 80^\circ \)

Since \( \overline{AB} \parallel \overline{CD} \), and \( \overline{EF} \) is the transversal, the sum of the angles around point \( E \) must form a straight line. This gives us the following equation:
\[
m\angle GHE + m\angle EGH + m\angle EFD = 180^\circ
\]

#### Step 3: Substitute the known values
\[
80^\circ + 40^\circ + x = 180^\circ
\]

#### Step 4: Solve for \( x \)
\[
120^\circ + x = 180^\circ
\]
\[
x = 180^\circ - 120^\circ
\]
\[
x = 60^\circ
\]

Thus, the value of \( x \) is \( \boxed{60^\circ} \).

---

Would you like further details or have any questions on this problem? Here are some related questions to consider:

1. How do alternate interior angles help in parallel line problems?
2. What is the sum of angles formed by a transversal intersecting parallel lines?
3. Can we apply any triangle angle-sum properties to this problem?
4. What happens to the angles when lines are parallel and a transversal crosses them?
5. How does the relationship between \( \overline{EF} \) and the parallel lines affect the angle calculations?

**Tip:** In geometry problems involving parallel lines and transversals, always look for alternate interior angles and corresponding angles, as they often provide key relationships to solve for unknowns.

In the accompanying diagram, AB ∥ CD, EF intersects AB at E and CD at F, and GH intersects AB at G and EF at H. If m∠EGH = 40°, m∠GHE = 80°, and m∠EFD = x, what is the value of x?

Solve a geometry problem involving parallel lines AB ∥ CD and a transversal EF, with intersecting angles. Learn how to calculate the unknown angle x using angle relationships and the alternate interior angles theorem. Suitable for students in Grades 9-11.

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