To solve for angles $x$ and $y$ in the given diagram, we need to use the geometrical properties of triangles, straight lines, and the sum of angles around a point. Here's the step-by-step solution:

Part (a): Angle $x$

1. Triangle ABC:
   • The sum of angles in a triangle is $180^\circ$.
   • The given angles in $\triangle ABC$ are $35^\circ$ and $85^\circ$.
   • To find $x$, calculate: $x = 180^\circ - 35^\circ - 85^\circ$ $x = 60^\circ$

Part (b): Angle $y$

2. Straight Line DE:
   • Angle $y$ and the $85^\circ$ angle on $\triangle ADE$ form a straight line at $A$.
   • The sum of angles on a straight line is $180^\circ$.
   • To find $y$, calculate: $y = 180^\circ - 85^\circ$ $y = 95^\circ$

Final Answers:

• $\mathbf{x = 60^\circ}$
• $\mathbf{y = 95^\circ}$

Do you want further clarifications or a more detailed breakdown of the steps?

Here are 5 related questions to expand your understanding:

1. How can you prove the sum of angles in a triangle is $180^\circ$?
2. What is the relationship between angles on a straight line?
3. How do you calculate unknown angles in quadrilaterals?
4. Can the same properties apply to non-Euclidean geometry?
5. What other methods can you use to verify angle measurements?

Tip: Always label your diagram carefully and check for supplementary and complementary angles in problems involving straight lines and triangles.