https://file.aiquickdraw.com/mathbot/file/1727173284872l12phbqr.jpg

In this image, we are tasked with finding the missing angles \( w^\circ \), \( x^\circ \), and \( y^\circ \) in a diagram featuring intersecting lines and labeled angles. Here's how we can approach it:

### Step-by-step solution:

1. **Finding angle \( w^\circ \)**:
   - Angle \( w^\circ \) forms a linear pair with the given \( 75^\circ \) angle (they are adjacent on a straight line).
   - Since the sum of angles on a straight line is always \( 180^\circ \), we can write:
     \[
     w + 75 = 180
     \]
     Solving for \( w \):
     \[
     w = 180 - 75 = 105^\circ
     \]

2. **Finding angle \( x^\circ \)**:
   - The angle \( x^\circ \) is vertically opposite to the given \( 84^\circ \) angle. Vertically opposite angles are equal, so:
     \[
     x = 84^\circ
     \]

3. **Finding angle \( y^\circ \)**:
   - Angle \( y^\circ \) forms a linear pair with the given \( 46^\circ \) angle (again, they are adjacent on a straight line).
   - Using the same reasoning as before:
     \[
     y + 46 = 180
     \]
     Solving for \( y \):
     \[
     y = 180 - 46 = 134^\circ
     \]

### Summary of Results:
- \( w = 105^\circ \)
- \( x = 84^\circ \)
- \( y = 134^\circ \)

Would you like further details on any of these steps or have any questions?

Here are some related questions:
1. What is the sum of angles in a triangle, and how does it relate to this problem?
2. How do you calculate vertically opposite angles?
3. What are linear pairs of angles, and how do they help in solving for missing angles?
4. How can alternate interior angles be used to solve for angles in parallel line diagrams?
5. Can corresponding angles help find missing angles in similar diagrams?

**Tip**: When dealing with intersecting lines, always check for vertical angles and linear pairs to quickly identify angle measures.

Find the values of the missing angles in the diagram below (involving intersecting lines and angles: 20°, 46°, 75°, and 84°).

In this problem, we find the missing angles w°, x°, and y° by using concepts of linear pairs and vertically opposite angles. Learn how to solve it step-by-step using key geometric theorems and relationships.

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