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This problem involves a pair of vertically opposite angles. Vertically opposite angles are always equal when two lines intersect.

Given the angles:

\[
(7z + 7)^\circ \quad \text{and} \quad (9z - 7)^\circ
\]

Since these are vertically opposite angles, we can set them equal to each other:

\[
7z + 7 = 9z - 7
\]

### Step-by-Step Solution:

1. First, subtract \(7z\) from both sides:

\[
7 = 2z - 7
\]

2. Next, add \(7\) to both sides:

\[
14 = 2z
\]

3. Finally, divide by \(2\) to solve for \(z\):

\[
z = 7
\]

Thus, the value of \(z\) is \(7\).

Would you like any further details or explanations?

Here are 5 related questions you might find interesting:

1. How can we prove that vertically opposite angles are always equal?
2. What are corresponding angles, and how do they relate to parallel lines?
3. How do supplementary angles differ from vertically opposite angles?
4. What happens to the angle values when lines are not intersecting at right angles?
5. Can this concept of vertically opposite angles be extended to three-dimensional geometry?

**Tip**: Always check whether angles are vertically opposite before assuming equality in intersecting line problems!

Solve for the value of z, given the vertically opposite angles (7z + 7)° and (9z - 7)°.

Learn how to solve for the variable z in a pair of vertically opposite angles where (7z + 7)° equals (9z - 7)°. This problem covers key algebra and geometry concepts, suitable for Grades 7-9.

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