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Let's go through each part of the problem, based on the given figure and the questions:

1. **The relation between \( x \) and \( y \)**:
   - In the figure, the road and railroad tracks form parallel lines, and \( x \) and \( y \) are angles formed by a transversal line (the road). Since \( x \) and \( y \) are on opposite sides of the transversal, but inside the parallel lines, they are **alternate interior angles**.
   - **Answer: A. Alternate interior angle**

2. **The angle between the angle bisectors of \( p \) and \( x \)**:
   - The bisector of an angle divides the angle into two equal parts. If the bisectors of \( p \) and \( x \) form an angle, it would be half the sum of the angles \( p \) and \( x \). Since \( p \) and \( x \) are supplementary angles (they add up to 180°), their bisectors will form an angle of \( 90^\circ \).
   - **Answer: D. 90°**

3. **Find the values of \( y \) and \( p \)**:
   - From the given figure, we know that \( y = x \) since they are alternate interior angles. The angle \( m \) is given as \( 50^\circ \). Therefore:
     - \( x + 50^\circ = 90^\circ \) (since \( x \) and 50° are complementary angles).
     - \( x = 40^\circ \).
     - Since \( x \) and \( y \) are equal, \( y = 40^\circ \).
   - \( p \) is supplementary to \( x \), so \( p + x = 180^\circ \). Thus, \( p = 180^\circ - 40^\circ = 140^\circ \).
   - **Values: \( y = 40^\circ \), \( p = 140^\circ \)**.

4. **Find the values of \( x \) and \( m \)**:
   - We already found that \( x = 40^\circ \) and \( m = 50^\circ \).
   - **Values: \( x = 40^\circ \), \( m = 50^\circ \)**.

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Let me know if you need further clarification or details. Here are five related questions to deepen your understanding:

1. What is the sum of the interior angles on the same side of a transversal cutting parallel lines?
2. Can you explain why alternate interior angles are equal in parallel lines?
3. What is the relationship between corresponding angles when a transversal cuts two parallel lines?
4. How do angle bisectors behave when the angles are supplementary?
5. What would happen to the values of \( p \) and \( y \) if the given angle was different?

**Tip:** Always remember that when dealing with parallel lines and a transversal, alternate interior angles are equal, and co-interior angles sum up to 180°.

In the figure, the road crossing the railroad tracks forms parallel lines & transversal. One corner side the angle formed between the road and the rail track is given as 50°. Find the relationship between x and y, the angle between the bisectors of p and x, and the values of y, p, x, and m.

Explore how to solve a geometry problem involving angles formed by a transversal crossing parallel lines. Learn about alternate interior angles, supplementary angles, and angle bisectors. Suitable for Grades 8-10 students.

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