https://file.aiquickdraw.com/mathbot/file/1725372418932z0hj5lmr.png

To solve for the missing angles \( x \) and \( y \) in the diagram, let's analyze the relationships between the angles.

1. **Given Information**:
   - The angle marked as \( 31^\circ \).
   - There is a right angle \( 90^\circ \) between the two lines, indicated by the square symbol.

2. **Relationship between Angles**:
   - The angle \( x \) and the angle \( 31^\circ \) are complementary since they are adjacent and form a right angle. Therefore:
     \[
     x + 31^\circ = 90^\circ
     \]
     \[
     x = 90^\circ - 31^\circ = 59^\circ
     \]

   - The angle \( y \) is on a straight line with the right angle and angle \( 31^\circ \). Angles on a straight line add up to \( 180^\circ \). Therefore:
     \[
     y + 90^\circ = 180^\circ
     \]
     \[
     y = 180^\circ - 90^\circ = 90^\circ
     \]

3. **Final Answers**:
   - \( x = 59^\circ \)
   - \( y = 90^\circ \)

Would you like more details on the steps or explanations? Here are some related questions to explore further:

1. How do complementary and supplementary angles differ?
2. What are the properties of angles formed by intersecting lines?
3. How do you determine if two angles are adjacent?
4. Can you explain how to solve for unknown angles in more complex diagrams?
5. What is the significance of the square symbol indicating a right angle in geometric problems?

**Tip:** Always look for right angles and straight lines in diagrams, as they provide key information for solving angle problems.

Find the measure of the missing angles x and y in the given diagram.

This problem involves solving for missing angles x and y in a geometric diagram. Learn how to use the Complementary Angles Theorem and the Supplementary Angles Theorem to find the correct values of x and y. Suitable for students in Grades 7-9.

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