This worksheet helps students practice identifying vertically opposite and adjacent angles formed by intersecting lines.

Problem Analysis and Solution

The task involves identifying and analyzing angles related to intersecting lines. Let's solve each part step by step.

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#### Part A: Identifying Vertically Opposite Angles and Adjacent Angles

Given:
- The diagram consists of four straight lines.
- We need to identify all pairs of vertically opposite angles and adjacent angles.

Diagram:
- The diagram shows two sets of intersecting lines:
1. Lines intersecting at point \( P \) with angles labeled \( p, q, z, y \).
2. Lines intersecting at point \( x \) with angles labeled \( r, s, w, x \).

Solution:

1. Vertically Opposite Angles:
- Vertically opposite angles are the angles opposite each other when two lines intersect. They are always equal.
- At point \( P \):
- \( p \) and \( q \)
- \( z \) and \( y \)
- At point \( x \):
- \( r \) and \( s \)
- \( w \) and \( x \)

2. Adjacent Angles:
- Adjacent angles are angles that share a common vertex and a common side but do not overlap.
- At point \( P \):
- \( p \) and \( z \)
- \( q \) and \( y \)
- \( p \) and \( y \)
- \( q \) and \( z \)
- At point \( x \):
- \( r \) and \( w \)
- \( s \) and \( x \)
- \( r \) and \( x \)
- \( s \) and \( w \)

Final Answer for Part A:
- Vertically Opposite Angles:
- \( p \) and \( q \)
- \( z \) and \( y \)
- \( r \) and \( s \)
- \( w \) and \( x \)
- Adjacent Angles:
- \( p \) and \( z \)
- \( q \) and \( y \)
- \( p \) and \( y \)
- \( q \) and \( z \)
- \( r \) and \( w \)
- \( s \) and \( x \)
- \( r \) and \( x \)
- \( s \) and \( w \)

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#### Part B: Marking Correct and Incorrect Statements

Given:
- JK, LM, and PQ are straight lines.
- We need to mark statements as correct (\(\checkmark\)) or incorrect (\(X\)) based on the diagram.

Diagram:
- The diagram shows three intersecting lines forming various angles:
- \( a, b, c, d, e \) are labeled around the intersections.

Statements:
1. \( b = e \)
2. \( d = 90^\circ \)
3. \( c = d \)

Solution:

1. Statement 1: \( b = e \)
- \( b \) and \( e \) are vertically opposite angles. Vertically opposite angles are always equal.
- Therefore, \( b = e \) is correct.

2. Statement 2: \( d = 90^\circ \)
- The diagram shows a right angle symbol at angle \( d \), indicating that \( d \) is a right angle.
- Therefore, \( d = 90^\circ \) is correct.

3. Statement 3: \( c = d \)
- \( c \) and \( d \) are adjacent angles formed by the intersection of lines. They are not equal unless additional information specifies otherwise.
- From the diagram, there is no indication that \( c \) and \( d \) are equal.
- Therefore, \( c = d \) is incorrect.

Final Answer for Part B:
1. \( b = e \) — \(\checkmark\)
2. \( d = 90^\circ \) — \(\checkmark\)
3. \( c = d \) — \(X\)

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#### Part C: Marking and Labeling the Angle \( y \)

Given:
- Three scenarios where \( x \) and \( y \) are described as either vertically opposite angles or adjacent angles.
- We need to mark and label the angle \( y \) accordingly.

Scenarios:
1. \( x \) and \( y \) are vertically opposite angles.
2. \( x \) and \( y \) are adjacent angles.
3. \( x \) and \( y \) are adjacent angles.

Solution:

1. Scenario 1: \( x \) and \( y \) are vertically opposite angles.
- Vertically opposite angles are opposite each other when two lines intersect.
- In the given diagram, \( y \) should be marked directly opposite \( x \).

2. Scenario 2: \( x \) and \( y \) are adjacent angles.
- Adjacent angles share a common vertex and a common side.
- In the given diagram, \( y \) should be marked next to \( x \) sharing a common side.

3. Scenario 3: \( x \) and \( y \) are adjacent angles.
- Similar to Scenario 2, \( y \) should be marked next to \( x \) sharing a common side.

Final Answer for Part C:
- Scenario 1: Mark \( y \) directly opposite \( x \).
- Scenario 2: Mark \( y \) next to \( x \) sharing a common side.
- Scenario 3: Mark \( y \) next to \( x \) sharing a common side.

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Final Answers:

- Part A:
- Vertically Opposite Angles: \( p \) and \( q \), \( z \) and \( y \), \( r \) and \( s \), \( w \) and \( x \)
- Adjacent Angles: \( p \) and \( z \), \( q \) and \( y \), \( p \) and \( y \), \( q \) and \( z \), \( r \) and \( w \), \( s \) and \( x \), \( r \) and \( x \), \( s \) and \( w \)

- Part B:
- 1. \( b = e \) — \(\checkmark\)
- 2. \( d = 90^\circ \) — \(\checkmark\)
- 3. \( c = d \) — \(X\)

- Part C:
- Scenario 1: Mark \( y \) directly opposite \( x \).
- Scenario 2: Mark \( y \) next to \( x \) sharing a common side.
- Scenario 3: Mark \( y \) next to \( x \) sharing a common side.

Boxed Final Answer:
\[
\boxed{
\text{Part A: Vertically Opposite: } (p, q), (z, y), (r, s), (w, x); \text{ Adjacent: } (p, z), (q, y), (p, y), (q, z), (r, w), (s, x), (r, x), (s, w) \\
\text{Part B: } 1. \checkmark, 2. \checkmark, 3. X \\
\text{Part C: See descriptions above.}
}
\]