https://file.aiquickdraw.com/mathbot/file/1725604594778ugp1wufr.jpg

In the image, we see two parallel lines \( AB \) and \( CD \), and a transversal line \( EF \) that cuts across them. The angle given is \( 74^\circ \) at the intersection of lines \( AB \) and \( EF \). We are tasked with finding angles \( x \), \( y \), and \( z \) formed by the intersections.

### Step-by-step analysis:
1. **Angle \( z \):**
   Angle \( z \) is the corresponding angle to the \( 74^\circ \) angle, since lines \( AB \) and \( CD \) are parallel and \( EF \) is the transversal. By the **corresponding angles postulate**, we have:
   \[
   z = 74^\circ
   \]

2. **Angle \( y \):**
   Angle \( y \) is a supplementary angle to the \( 74^\circ \) angle because they are on a straight line (i.e., they are **linear pairs**). Therefore, we calculate \( y \) as:
   \[
   y = 180^\circ - 74^\circ = 106^\circ
   \]

3. **Angle \( x \):**
   Angle \( x \) is a corresponding angle to \( y \) because lines \( AB \) and \( CD \) are parallel. By the **corresponding angles postulate**, we have:
   \[
   x = y = 106^\circ
   \]

### Summary:
- \( z = 74^\circ \)
- \( y = 106^\circ \)
- \( x = 106^\circ \)

Would you like more details on any of these steps, or do you have other questions?

Here are some related questions:
1. What are alternate interior angles and how do they relate to transversal lines?
2. How can we verify whether two lines are parallel using transversal angles?
3. What is the sum of interior angles on the same side of the transversal for two parallel lines?
4. Can you explain the difference between corresponding angles and alternate angles?
5. How would the angle relationships change if the lines were not parallel?

**Tip:** When two lines are cut by a transversal, corresponding angles are equal if the lines are parallel.

Given two parallel lines AB and CD, intersected by a transversal EF, with the angle at the intersection of AB and EF given as 74°, find angles x, y, and z.

Learn how to find the angles x, y, and z when two parallel lines are intersected by a transversal with a given angle of 74°. This problem demonstrates the use of corresponding angles and linear pair relationships in geometry, suitable for Grades 8-10.

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