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To solve for the missing angles \( w^\circ \), \( x^\circ \), \( y^\circ \), and \( z^\circ \), let's analyze the diagram step by step:

### Step 1: Identifying Relationships between the Angles
- The angle labeled \( 76^\circ \) and \( w^\circ \) are **vertically opposite angles**, meaning they are equal.
  \[
  w = 76^\circ
  \]

- The angle labeled \( 39^\circ \) is **alternate interior** to \( x^\circ \) (since the two lines are parallel), meaning they are also equal.
  \[
  x = 39^\circ
  \]

### Step 2: Using the Triangle Angle Sum Theorem
In the triangle formed by \( x^\circ \), \( y^\circ \), and the \( 97^\circ \) angle:
- The sum of angles in a triangle is always \( 180^\circ \).
\[
x + y + 97^\circ = 180^\circ
\]
Substitute \( x = 39^\circ \):
\[
39^\circ + y + 97^\circ = 180^\circ
\]
\[
y = 180^\circ - (39^\circ + 97^\circ) = 180^\circ - 136^\circ = 44^\circ
\]

### Step 3: Finding \( z^\circ \)
The angle labeled \( z^\circ \) is supplementary to the angle \( 97^\circ \) (since they are on a straight line). Therefore:
\[
z + 97^\circ = 180^\circ
\]
\[
z = 180^\circ - 97^\circ = 83^\circ
\]

### Final Results:
- \( w = 76^\circ \)
- \( x = 39^\circ \)
- \( y = 44^\circ \)
- \( z = 83^\circ \)

Let me know if you would like further details or have any questions. Here are some related questions:

1. How do vertically opposite angles help solve for unknown angles?
2. What is the significance of alternate interior angles in parallel lines?
3. Can you apply the triangle angle sum theorem to any other triangle in the diagram?
4. How do supplementary angles relate to finding unknown angles on a straight line?
5. What would change if the lines in the diagram weren't parallel?

**Tip**: Always look for parallel lines in geometric problems to leverage alternate interior and corresponding angle relationships!

Find the values of the missing angles in the diagram.

Learn how to find the missing angles in a diagram involving parallel lines and triangles. This solution applies key geometric concepts such as the Triangle Angle Sum Theorem, alternate interior angles, and vertically opposite angles. Suitable for students in Grades 8-10.

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