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Let's solve the problem step by step based on the provided geometrical figure and details:

### Given:
- \( AB \parallel CD \parallel GF \) and \( AG \perp AB \).
- The diagram shows multiple parallel lines and transversal lines intersecting these parallels.

### i) **Find the measure of angle \( x \)**:
- \( \angle G = 70^\circ \) is given in the diagram.
- Since \( AB \parallel GF \) and \( AG \) is the transversal, angle \( x \) is corresponding to \( \angle G = 70^\circ \) because corresponding angles are equal when lines are parallel.

Thus, \( x = 70^\circ \).

### ii) **Find the value of \( y \)**:
- \( AB \parallel CD \) and \( AG \) is a transversal. Therefore, \( \angle D + \angle G = 180^\circ \) (co-interior angles on the same side of the transversal are supplementary).
- Given \( \angle G = 70^\circ \), we have:
  \[
  y + 70^\circ = 180^\circ
  \]
  Solving for \( y \):
  \[
  y = 110^\circ
  \]

### OR (Alternative option):
If \( z = y \), find the value of \( x + y + z \).

Since \( z = y \), we already know \( y = 110^\circ \). Therefore, \( z = 110^\circ \).

Now, calculate \( x + y + z \):
\[
x + y + z = 70^\circ + 110^\circ + 110^\circ = 290^\circ
\]

### iii) **Find the measure of \( \angle BDC \)**:
- From the diagram, \( \angle BDC \) is formed by the transversal \( BD \) crossing parallel lines \( AB \) and \( CD \).
- Since \( AB \parallel CD \), \( \angle BDC \) is equal to \( \angle ACD \) (alternate interior angles). Hence, \( \angle BDC = x = 70^\circ \).

### Summary:
1. \( x = 70^\circ \).
2. \( y = 110^\circ \).
3. \( x + y + z = 290^\circ \) (if \( z = y \)).
4. \( \angle BDC = 70^\circ \).

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Would you like further details or explanations on any part of the solution?

Here are 5 related questions to expand on this:
1. How do alternate interior angles work when lines are parallel?
2. Can you explain co-interior angles in parallel lines and transversal intersections?
3. Why are corresponding angles equal when a transversal crosses parallel lines?
4. What is the difference between alternate and corresponding angles?
5. How can you prove lines are parallel using angle relationships?

**Tip**: Always check for parallel lines and transversal intersections to apply angle theorems effectively.

The original question submitted by the user as text.

Explore how to solve problems involving parallel lines and transversals with detailed solutions for angle calculations using corresponding angles, co-interior angles, and alternate interior angles theorems. This guide is suitable for students in Grades 8-10, enhancing understanding of fundamental geometry concepts.

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