The image you uploaded shows a figure with intersecting lines and some angles labeled. The problem asks to find the values of $x$ and $z$.

Explanation:

Given:

• The angle opposite to $78^\circ$ is also $78^\circ$ (vertical angles are equal).
• The angle opposite to $z^\circ$ is $z^\circ$ (vertical angles are equal).
• The angle $(10x + 58)^\circ$ is adjacent to $78^\circ$ and $z^\circ$.

Steps to solve:

1. Use the property of vertical angles: $\text{Angle opposite } 78^\circ = 78^\circ$ $\text{Angle opposite } z^\circ = z^\circ$

2. Use the linear pair (angles on a straight line) property: The sum of adjacent angles on a straight line is $180^\circ$. $78^\circ + (10x + 58)^\circ = 180^\circ$

3. Solve the equation for $x$: $78 + 10x + 58 = 180$ Combine like terms: $10x + 136 = 180$ Subtract 136 from both sides: $10x = 44$ Divide by 10: $x = 4.4$

4. Determine $z$: Since $z$ is vertical to $78^\circ$, and these are vertical angles: $z = 78^\circ$

Final Answers:

• $x = 4.4$
• $z = 78^\circ$

Would you like a more detailed explanation or have any questions about these steps?

Here are five related questions that could further your understanding:

1. What is the linear pair of angles and how does it apply to this problem?
2. How do you determine vertical angles, and why are they equal?
3. Can $x$ take any value other than $4.4$ to satisfy the equation?
4. What are supplementary angles, and how do they differ from complementary angles?
5. How would the solution change if the angle opposite $z^\circ$ was given as a different value?

Tip: Remember, vertical angles are always equal, and angles on a straight line always add up to 180°.