In this image, we are given two parallel lines $k \parallel n$ and a transversal line cutting through them, forming several angles. The task is to find the values of $x$ and $z$, given:

• $x^\circ$ as an unknown angle,
• $(6z - 90)^\circ$ as an expression for an angle,
• One known angle, $66^\circ$.

Step 1: Identify relationships between angles

Since lines $k \parallel n$ are parallel, we can apply the angle properties formed by a transversal:

• Corresponding angles: Angles that are on the same side of the transversal and in the same relative position.
• Alternate interior angles: Angles that are inside the parallel lines and on opposite sides of the transversal.

From the figure:

• The angle labeled $66^\circ$ and the angle $x^\circ$ are corresponding angles. So, by the corresponding angles postulate: $x = 66^\circ$

Step 2: Solve for $z$

The angle expression $(6z - 90)^\circ$ is vertically opposite to the angle labeled $x^\circ$, and vertically opposite angles are equal. Therefore: $6z - 90 = 66$ Solve for $z$: $6z = 66 + 90$ $6z = 156$ $z = \frac{156}{6}$ $z = 26$

Final Answer:

• $x = 66^\circ$
• $z = 26$

Would you like further explanation or details on the angle properties used?

Here are 5 related questions for you:

1. What are corresponding angles and how are they identified?
2. How do alternate interior angles behave when a transversal cuts parallel lines?
3. Could vertical angles ever differ from one another in measure?
4. What happens to the sum of angles on a straight line?
5. Why are parallel lines important in geometry?

Tip: Always look for known angle relationships (like corresponding or alternate interior angles) when dealing with parallel lines and transversals.