Identify Adjacent Angles Formed by Parallel Lines and Transversal

Published on March 26, 2026

Geometry

The problem asks to identify which pair of angles shares a common vertex and a common side from the figure involving two parallel lines cut by a transversal. The correct pair forms a linear pair at one of the intersection points.

Question

Answer

Here is the step-by-step solution to find the pair of adjacent angles.

1. Definition of Adjacent Angles

Two angles are adjacent if they share a common vertex and a common side (ray), but do not overlap (i.e., their interiors are disjoint).

In the context of the diagram, the angles must meet at the same point and share one of the rays forming the angle.

2. Analyze the Intersections and Angles

The transversal line

intersects the parallel lines

and

at points

and

, respectively.

Angles at vertex K: , , , (or )
Angles at vertex N: , (or ), (or ), (or )

3. Evaluate the Options

We check each pair to see if they share a common vertex and a common side.

Option 1: and
- Vertex of is .
- Vertex of is .
- They have different vertices (). Therefore, they are not adjacent. (These are alternate interior angles, which are equal because the lines are parallel, but not adjacent).
Option 2: and
- Vertex of is .
- Vertex of is .
- They have different vertices (). Therefore, they are not adjacent. (These are corresponding angles, which are equal, but not adjacent).
Option 3: and
- Vertex of is .
- Vertex of is . (They share the common vertex ).
- The sides of are ray and ray .
- The sides of are ray and ray .
- Wait, is the angle formed by ray and ray . The angle is the straight angle along line on the side of . Let’s re-examine the angles formed at by the intersection of and .
- The four angles formed at are: , , (or ), and (or ).
- The pair and :
  - uses rays and .
  - uses rays and (where is on the ray ). Thus is the same as .
- and are adjacent. and form a linear pair on line , so they are adjacent.
- Let’s assume the notation refers to the angle formed by the ray and the ray . Since are collinear, is the same ray as . If , then and would be the same angle. This seems unlikely for a multiple-choice question unless the diagram implies are points on the transversal.
- Let’s re-read the standard naming convention for angles formed by two intersecting lines, and . The angles around are , , , and .
- If we consider , since is on the ray , then is the same as .
- If we consider and :
  - They share vertex .
  - uses rays and .
  - uses rays and .
  - Since and are opposite rays (they form the line ), and form a linear pair, and thus are adjacent. (They share side if and are considered, but here they share the line ).
- If means , then and are adjacent angles forming a linear pair along the transversal (with ray as the shared side between and , and and forming a linear pair along ).
- Let’s stick to the most direct definition: Two angles are adjacent if they share a ray and a vertex.
- and are adjacent (share vertex , share ray ).
- and are adjacent (share vertex , share ray ).
- In option 3: and (which is ). These angles are supplementary (form a linear pair along line ), meaning they are adjacent. They share the vertex . They share the ray if we consider to be , which it is not. They do not share a common side, they are separated by the line (unless we consider the rays involved: is the side of , and is the side of . Since and are opposite rays, they do not share a side). Wait, this implies they are not adjacent based on the strict definition.
- Let’s re-examine the visual relationships for adjacency: Angles that form a linear pair are adjacent.
- and form a straight line ? No, is a straight angle. and form the straight angle . Thus, and are adjacent angles forming a linear pair along line .
- and are adjacent angles forming a linear pair along line .
- Let’s check the pairs again focusing on shared rays:
  - If Option 3 meant and , they are the same angle, not distinct adjacent angles.
  - If Option 3 meant and : They share vertex and ray . They are adjacent.
  - If Option 3 meant and : They share vertex and ray . They are adjacent.
- Given the options, let’s reconsider . Since are collinear, . and are supplementary because is a side for both, and and are opposite rays. Thus, and are adjacent.
- Conclusion for Option 3: and () share vertex . They are separated by the ray ? No, they are separated by the ray ( are opposite). They form a linear pair along line if they shared the ray . They are adjacent because they form a linear pair along line if we consider and .
- Let’s assume the angle pairing that forms a linear pair (which are always adjacent) is intended here. and form a linear pair along line . and form a linear pair along line . is . and are supplementary (add to ) and share vertex . They are adjacent because they share the side . Wait, uses and uses and . Let’s look closer at the rays forming and .
  - :
  - :
  - Since and are opposite rays, the union of the two angles () does not form a angle unless are collinear, which they are. The union forms , a straight angle (). For two angles to be adjacent and form a linear pair, they must share one ray, and the other two rays must form a straight line.
  - If they share ray , the other rays must be and . If and are opposite rays, they are adjacent and form a linear pair. and are indeed opposite rays because form a line.
  - Therefore, and are adjacent and form a linear pair along line .
  - The option is and (). This is not the linear pair . Thus, Option 3 is unlikely based on the strict definition unless is a typo for .
Option 4: and
- Vertex: Both angles share vertex .
- Common Side: Both angles share the ray .
- Non-overlapping: The other rays, and , are opposite rays because form a straight line .
- Since they share a vertex and a side, and their non-shared sides form a straight line, and are adjacent angles (and form a linear pair).