https://file.aiquickdraw.com/mathbot/file/17302140677312dq5ge86.jpg

To solve for \( x \) and \( z \) in this figure, we will use the properties of vertically opposite angles.

1. The expressions \( (10x + 68)^\circ \) and \( (13x + 53)^\circ \) represent two vertically opposite angles, which are equal. Thus, we set up the equation:
   \[
   10x + 68 = 13x + 53
   \]

2. Rearrange to solve for \( x \):
   \[
   68 - 53 = 13x - 10x
   \]
   \[
   15 = 3x
   \]
   \[
   x = 5
   \]

3. Now, substitute \( x = 5 \) back into one of the angle expressions (for example, \( 10x + 68 \)) to find the angle value:
   \[
   10(5) + 68 = 50 + 68 = 118^\circ
   \]

4. Since vertically opposite angles are equal, both \( (10x + 68)^\circ \) and \( (13x + 53)^\circ \) equal \( 118^\circ \).

5. For the value of \( z \), observe that \( z^\circ \) is the supplementary angle to \( 118^\circ \) (since they form a straight line), so:
   \[
   z = 180^\circ - 118^\circ = 62^\circ
   \]

### Final answers:
- \( x = 5 \)
- \( z = 62^\circ \)

Would you like further details or have any questions?

Here are some related questions for additional practice:

1. What is the sum of angles around a point?
2. How do you identify vertically opposite angles?
3. How can you solve for unknowns in equations involving angles?
4. What are supplementary angles, and how are they related?
5. How does rearranging equations help in solving for unknowns?

**Tip:** Always remember that vertically opposite angles are equal. This property is particularly useful when working with intersecting lines.

Given the figure below, find the values of x and z.

Find the values of x and z by using the properties of vertically opposite angles and supplementary angles. This problem involves setting up equations and solving for unknowns, suitable for students in Grades 8-10.

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