https://file.aiquickdraw.com/mathbot/file/1726194102600zvz2nhkm.jpg

In the given image, we are tasked with determining the measures of the unknown angles in a geometric figure. Here's the approach:

1. **Identify given angles**:
   - One of the angles is marked as \(135^\circ\).
   - Another angle is marked as \(61^\circ\).

2. **Use the linear pair property**:
   The angle next to the \(135^\circ\) angle is marked as \(y^\circ\). Since the two form a straight line, we can use the fact that linear pairs sum to \(180^\circ\).

   Therefore:
   \[
   y + 135^\circ = 180^\circ
   \]
   Solving for \(y\):
   \[
   y = 180^\circ - 135^\circ = 45^\circ
   \]

3. **Use the triangle angle sum property**:
   Now, for the triangle containing angles \(x^\circ\), \(61^\circ\), and \(y = 45^\circ\), we know that the sum of the interior angles of a triangle is always \(180^\circ\).

   Therefore:
   \[
   x + 61^\circ + 45^\circ = 180^\circ
   \]
   Simplifying:
   \[
   x + 106^\circ = 180^\circ
   \]
   Solving for \(x\):
   \[
   x = 180^\circ - 106^\circ = 74^\circ
   \]

### Summary of Results:
- \(x = 74^\circ\)
- \(y = 45^\circ\)

Would you like any further details or have any questions?

### Related questions:
1. What are alternate interior angles, and how can they help solve angle problems?
2. How can the exterior angle theorem help find unknown angles in triangles?
3. What is the sum of interior angles in polygons other than triangles?
4. How are complementary and supplementary angles used in geometry?
5. Can the properties of parallel lines help in solving more complex angle problems?

### Tip:
When solving for unknown angles in a triangle, always remember that the sum of the interior angles is \(180^\circ\), which can be a useful shortcut!

Determine the measures of the unknown angles in the figure provided, where one angle is marked as 61°, another is 135°, and the remaining two angles are marked as x° and y°.

Learn how to find unknown angles in a geometric figure using the Linear Pair Theorem and the Triangle Angle Sum Theorem. This solution involves calculating the angles x and y given 135° and 61° angles. Suitable for middle school students in Grades 6-8.

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