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Let's analyze the given geometry problem step by step:

- We are given that:
  - Line **BA** is parallel to line **GH**.
  - Line **BC** is perpendicular to line **CD**.
  - Several angle measures are provided:
    - \(\angle ABC = 40^\circ\),
    - \(\angle CDE = 100^\circ\),
    - \(\angle DEF = 70^\circ\),
    - \(\angle EFG = 30^\circ\).

- We need to find the value of \(\angle DGH = x^\circ\).

### Key Information:
1. **Parallel lines**: Since **BA** is parallel to **GH**, the angles formed with the transversal can help us calculate the required angle using properties of parallel lines.
   
2. **Perpendicularity**: **BC** is perpendicular to **CD**, which tells us that \(\angle BCD = 90^\circ\).

### Solution:

1. **Corresponding Angles with Parallel Lines**:
   Since **BA** is parallel to **GH**, the angles formed with transversals can give corresponding angles. The sum of all angles in a zigzag path formed by the vertices (i.e., \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\)) will add up to the external angle at the final point \(H\).

2. **Exterior Angle Sum**:
   The exterior angle theorem states that the exterior angle is equal to the sum of the non-adjacent interior angles. We will sum all the given angles along the zigzag path and find the final angle \(\angle DGH\).

The exterior angle at point \(G\), \(\angle DGH = x\), is related to the previous interior angles:
\[
x = \left( \angle ABC - \angle CDE + \angle DEF - \angle EFG \right)
\]
Substituting the given values:
\[
x = 40^\circ - 100^\circ + 70^\circ - 30^\circ = -60^\circ + 70^\circ - 30^\circ = 10^\circ
\]

Thus, the value of \(\angle DGH = x\) is \(\boxed{10^\circ}\).

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Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What are corresponding angles and how do they relate to parallel lines?
2. How do perpendicular lines influence the angle relationships in a figure?
3. Can the exterior angle theorem be used in other polygon shapes? How?
4. How does the concept of trans

[BA // GH, BC ⊥ CD, m(∠ABC) = 40°, m(∠CDE) = 100°, m(∠DEF) = 70°, m(∠EFG) = 30°, m(∠DGH) = x° ? What is x?

This problem explores angle relationships formed by parallel lines, perpendicular lines, and transversals. Given several angles, find the value of ∠DGH using the exterior angle theorem and corresponding angles postulate. Suitable for Grades 8-10 students studying geometry.

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