Use betweenness of Rays to answer this question.

Question

If point K lies in the interior of <LMN and m<KMN =30, m<LMN= 75, and m<LMK=45, then which of these rays is between the other two?

Answer 1

To determine which ray is between the other two based on the given angles, we can use the property of angles and their measurements.

Given:

• \( m \angle KMN = 30^\circ \)
• \( m \angle LMN = 75^\circ \)
• \( m \angle LMK = 45^\circ \)

1. Since \( K \) lies in the interior of \( \angle LMN \), we know that the measure of \( \angle LMN \) can be expressed as the sum of \( m \angle LMK \) and \( m \angle KMN \): \[ m \angle LMN = m \angle LMK + m \angle KMN. \] Substituting the values: \[ 75^\circ = 45^\circ + 30^\circ, \] which verifies that the angles add up correctly.

2. To determine which ray is between the other two, recall that \( \angle LMK \) is formed by ray \( LK \) and ray \( LM \), and angle \( KMN \) is formed by ray \( KM \) and ray \( KN \).

3. The ray \( LK \) has an angle measure of \( 45^\circ \), the ray \( KM \) has an angle measure of \( 30^\circ \), and ray \( LMN \) includes both \( LMK \) and \( KMN \).

Thus, we can conclude the relationship between the rays based on the angles:

• Since \( K \) is between \( L \) and \( M \) when we measure in the direction of \( L \) to \( M \), the angle \( KMN \) being the smallest angle at \( K \) implies that ray \( LK \) is between ray \( KM \) and ray \( LM \) because \( LMK \) (45°) is greater than \( KMN \) (30°).

The correct conclusion is that ray \( LK \) is between rays \( LM \) and \( KM \).

Answer 2

A. ray MK is between ML and MN
B. ray ML is between MK and MN
C. ray MN is between MK and ML

Answer 3

To determine which ray is between the other two using the information provided about angles, let's summarize the measures:

1. \( m \angle KMN = 30^\circ \)
2. \( m \angle LMN = 75^\circ \)
3. \( m \angle LMK = 45^\circ \)

In terms of the positioning of the rays relative to each other:

• Point \( K \) is in the interior of \( \angle LMN \).
• \( LMK \) (45°) is formed by ray \( LM \) and ray \( LK \).
• \( KMN \) (30°) is formed by ray \( KM \) and ray \( KN \).

To determine which ray is between the others, we must consider the measures of the angles. The relationships among them can help us understand their order:

• The measure of \( \angle LMK = 45^\circ \) is greater than \( \angle KMN = 30^\circ \) but less than \( \angle LMN = 75^\circ \). This implies that ray \( LK \) is between rays \( LM \) and \( KM \).

From the multiple-choice options:

A. ray MK is between ML and MN
B. ray ML is between MK and MN
C. ray MN is between MK and ML

Based on our previous analysis, we identified that ray LM is the one that forms the larger angle with ray LK compared to ray KM, confirming that:

None of the provided options directly fit the relationship we established. However, since we confirmed that ray \( LK \) (or \( ML \)) is between the other two rays (due to \( m \angle LMK = 45^\circ \) being greater than \( m \angle KMN = 30^\circ \)), the closest option from the list that aligns with the established order would have to involve ray \( ML \), making option B: ray ML is between MK and MN possibly the most relevant in the context of the angles provided.

In conclusion, the best answer from the options given is:

B. ray ML is between MK and MN.