Question

marcus states that angle orp and angle lrp are a linear pair. which best describes his statement? he is correct. the angles share a common vertex so they are a linear pair. he is correct. the angles share a common ray so they are a linear pair. he is incorrect. ray ro and ray rl are not opposite rays. he is incorrect. angle orp does not form a linear pair with another angle in the diagram.

Explanation:

Step1: Recall Linear Pair Definition

A linear pair of angles are adjacent angles (share a common side/ray) and their non - common sides form a straight line (supplementary, sum to \(180^\circ\)).

Step2: Analyze Angle ORP and Angle LRP

• For angle \(ORP\) and angle \(LRP\): Check if they share a common vertex and a common ray, and if their non - common rays form a straight line.
• The vertex of both angles is \(R\). The common ray is \(RP\). Now, check the non - common rays: For angle \(ORP\), non - common ray is \(RO\); for angle \(LRP\), non - common ray is \(RL\). \(RO\) and \(RL\) do not form a straight line (since \(RO\) is in a different direction from \(RL\) as seen in the diagram). So, angle \(ORP\) and angle \(LRP\) do not form a linear pair. So Marcus is incorrect.
• Now, check the options:
• Option 1: "He is incorrect. Angle ORP does not form a linear pair with another angle in the diagram." Let's verify. A linear pair requires two adjacent angles whose non - common sides are opposite rays (form a straight line). Angle \(ORP\) and angle \(NRP\) (if we consider \(RN\) opposite to \(RO\)? Wait, no, \(RO\) and \(RM\) (wait, in the diagram, \(RO\) is horizontal left, \(RP\) is vertical up, \(RN\) is vertical down, \(RM\) is horizontal right. Wait, angle \(ORP\) (between \(RO\) and \(RP\)) and angle \(NRP\) (between \(RN\) and \(RP\)): \(RO\) and \(RN\) are opposite rays (since \(RO\) is left horizontal, \(RN\) is down vertical? No, wait \(RO\) and \(RM\) are opposite rays (horizontal line), \(RP\) and \(RN\) are opposite rays (vertical line). So angle \(ORP\) (between \(RO\) and \(RP\)) and angle \(NRP\) (between \(RN\) and \(RP\)): \(RO\) and \(RN\) are not opposite rays. Wait, maybe angle \(ORP\) and angle \(NR O\)? No, let's get back to the options. The option "He is incorrect. Angle ORP does not form a linear pair with another angle in the diagram" is correct because for a linear pair, we need two angles. Since angle \(ORP\) does not have a second angle with which it can form a linear pair (because to form a linear pair with \(ORP\), the other angle would need to have a common ray \(RP\) and a non - common ray that is opposite to \(RO\). But in the diagram, there is no such angle. Wait, actually, angle \(ORP\) and angle \(NRP\): Wait, \(RO\) and \(RN\) are not opposite rays. \(RO\) is horizontal left, \(RN\) is vertical down. Opposite rays for \(RO\) would be \(RM\) (horizontal right), and opposite rays for \(RP\) would be \(RN\) (vertical down). So angle \(ORP\) (between \(RO\) and \(RP\)) and angle \(NRP\) (between \(RN\) and \(RP\)): \(RO\) and \(RN\) are not opposite rays, so they don't form a linear pair. Angle \(ORP\) and angle \(LRP\) also don't. So the option "He is incorrect. Angle ORP does not form a linear pair with another angle in the diagram" is the correct description of his statement.

Answer:

The option "He is incorrect. Angle ORP does not form a linear pair with another angle in the diagram." (the fourth option in the vertical list of options)