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. $\angle BDC = 70^\circ$.

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.