Let's solve the given questions step by step:

Given Information:

• $AB \parallel CD$
• $\angle A = 35^\circ$
• $\angle B = Z$
• $\angle O = 75^\circ$
• Angles $X, Y, W$ are to be calculated.

(a) Finding the measure of $\angle AOC$:

• $AB \parallel CD$ and $AO$ and $CO$ are transversals.
• $\angle A = 35^\circ$.
• By the property of alternate interior angles: $\angle AOC = \angle A + \angle O = 35^\circ + 75^\circ = 110^\circ.$

(b) Finding the measure of $\angle Y$:

• Using the property of a straight line: $\angle AOC + \angle Y = 180^\circ.$
• Substituting $\angle AOC = 110^\circ$: $\angle Y = 180^\circ - 110^\circ = 70^\circ.$

(c) Finding the measure of $\angle W$:

• Vertically opposite angles are equal: $\angle W = \angle O = 75^\circ.$

(d) Finding the measure of $\angle X$:

• Using the property of angles in a straight line $\angle W + \angle X = 180^\circ$: $\angle X = 180^\circ - \angle W = 180^\circ - 75^\circ = 105^\circ.$

Final Answers:

(a) $\angle AOC = 110^\circ$
(b) $\angle Y = 70^\circ$
(c) $\angle W = 75^\circ$
(d) $\angle X = 105^\circ$

Would you like further clarification on any step, or do you have any additional questions?

Related Questions to Explore:

1. What are alternate interior angles, and how are they used in parallel lines?
2. Why are vertically opposite angles equal?
3. How do you calculate angles on a straight line?
4. Can you prove why $AB \parallel CD$ affects the angles formed at $O$?
5. How can we verify this solution geometrically?

Tip:

Always double-check angle relationships (e.g., supplementary and vertically opposite) when dealing with parallel lines and transversals!