Let's analyze the problem.

The given diagram shows a set of angles formed by intersecting lines. We are tasked with finding the missing angles, labeled as $x$ and $y$.

Step-by-step solution:

1. Identify known angles:

   • One of the angles is labeled $30^\circ$.
   • The figure also shows a right angle ($90^\circ$) formed by two perpendicular lines.

2. Use the fact that straight angles measure $180^\circ$: The angle $x$ and the angle $30^\circ$ are adjacent and lie on a straight line. Therefore: $x + 30^\circ = 180^\circ$ Solving for $x$: $x = 180^\circ - 30^\circ = 150^\circ$

3. Use vertical angles: Angle $y$ is vertical to the right angle ($90^\circ$). Since vertical angles are equal, we know: $y = 90^\circ$

Final answer:

• $x = 150^\circ$
• $y = 90^\circ$

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How do vertical angles help solve problems involving intersecting lines?
2. What is the sum of angles on a straight line, and how is it useful in geometry?
3. How can you use the properties of perpendicular lines to find unknown angles?
4. Can supplementary angles always be used to find missing angle measures?
5. What role do adjacent angles play when working with straight-line diagrams?

Tip: When dealing with diagrams involving straight lines and angles, always check for relationships like vertical angles, straight angles, or right angles to simplify your calculations.