Question

angle klm and angle mln are a linear pair.

ray lr is to be added to the diagram so that it is opposite ray lm. which is true about angle rln that would be formed?

angles rln and mlk would be vertical angles.
angles rln and mln would be vertical angles.
angles rln and klm would be a linear pair.
angles rln and kln would be a linear pair.

Explanation:

Analyze the given geometric configuration

Using the Rays and Segments knowledge point, we identify the existing elements in the diagram:

• Points \(K\), \(L\), and \(N\) lie on a straight line, with \(L\) between \(K\) and \(N\). Thus, ray \(LK\) and ray \(LN\) are opposite rays.
• Ray \(LM\) shares the endpoint \(L\) and forms a linear pair of angles: \(\angle KLM\) and \(\angle MLN\).

Define the new ray and identify opposite rays

We are given that ray \(LR\) is added to the diagram such that it is opposite to ray \(LM\).

• Since ray \(LR\) is opposite to ray \(LM\), they form a straight line \(MR\).
• We already know that ray \(LK\) and ray \(LN\) are opposite rays, forming the straight line \(KN\).

Identify the relationship between the resulting angles

Using the Vertical Angles Theorem knowledge point, we analyze the intersecting lines:

• Line \(MR\) and line \(KN\) intersect at point \(L\).
• Two intersecting lines form two pairs of vertical angles.
• The pairs of opposite (vertical) angles are:

1. \(\angle RLN\) and \(\angle KLM\)
2. \(\angle RLK\) and \(\angle MLN\)

Evaluate the given multiple-choice options

• Option 1: "Angles RLN and MLK would be vertical angles." Since \(\angle MLK\) is the same angle as \(\angle KLM\), this statement is true.
• Option 2: "Angles RLN and MLN would be vertical angles." These angles share ray \(LN\) and are adjacent, not vertical.
• Option 3: "Angles RLN and KLM would be a linear pair." These are vertical angles, not a linear pair.
• Option 4: "Angles RLN and KLN would be a linear pair." \(\angle KLN\) is a straight angle (\(180^\circ\)), so they do not form a linear pair.

Answer:

• (A) Angles RLN and MLK would be vertical angles. (Correct answer)
• (B) Angles RLN and MLN would be vertical angles.
• (C) Angles RLN and KLM would be a linear pair.
• (D) Angles RLN and KLN would be a linear pair.