https://file.aiquickdraw.com/mathbot/file/17276657830808qtijhq3.jpg

In this problem, you are asked to find the values of \( x \) and \( z \) based on the figure, where lines \( o \parallel n \), meaning the two lines are parallel, and you are given the following angles:

- \( (6z - 95)^\circ \) on the top left.
- \( x^\circ \) on the bottom left.
- \( 77^\circ \) on the bottom right.

### Step 1: Using the parallel line properties
Since \( o \parallel n \), we can apply the properties of alternate interior angles and corresponding angles.

1. **Corresponding angles**:
   The angle marked \( 77^\circ \) corresponds to the angle \( x^\circ \) on the left because they are on the same side of the transversal and in corresponding positions. Therefore, we have:
   \[
   x = 77^\circ
   \]

### Step 2: Solving for \( z \)
The angles \( (6z - 95)^\circ \) and \( 77^\circ \) are **alternate interior angles**, which are equal because the lines are parallel. Thus, we set the two angles equal:
\[
6z - 95 = 77
\]

### Step 3: Solving the equation for \( z \)
Now, solve the equation:
\[
6z - 95 = 77
\]
Add 95 to both sides:
\[
6z = 172
\]
Now divide by 6:
\[
z = \frac{172}{6} = 28.67
\]

### Final answers:
- \( x = 77^\circ \)
- \( z \approx 28.67 \)

Would you like any additional details or clarification on the steps?

Here are 5 related questions for further exploration:
1. What are alternate interior angles, and why are they equal when lines are parallel?
2. How do corresponding angles help in solving geometry problems with parallel lines?
3. How would the solution change if the lines were not parallel?
4. Can you explain how vertical angles work in relation to parallel lines and a transversal?
5. What are the different types of angles formed when a transversal crosses two parallel lines?

**Tip**: When solving geometric problems involving parallel lines, always look for alternate interior and corresponding angles, as they often provide direct relationships.

In the figure below, o || n. Find the values of x and z. The given angles are (6z - 95)°, x°, and 77°.

This problem involves finding the values of x and z using properties of parallel lines and angles. Given the angles (6z - 95)°, x°, and 77°, the solution uses theorems such as the Corresponding Angles and Alternate Interior Angles to solve for x and z. Suitable for students in Grades 8-10.

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