To determine which ray is between the others, we can analyze the angles given.
- The measure of angle \( \angle KMN = 30^\circ \).
- The measure of angle \( \angle LMN = 75^\circ \).
- The measure of angle \( \angle LMK = 45^\circ \).
Since point K lies in the interior of angle \( \angle LMN \), we know that:
\[ m\angle LMN = m\angle LMK + m\angle KMN \]
Substituting the values we have:
\[ 75^\circ = 45^\circ + 30^\circ \]
This equation is true and confirms that the angles are consistent.
Next, we can determine which ray is between the others by looking at the angles:
- The angle \( \angle LMK \) is 45°.
- The angle \( \angle KMN \) is 30°.
Since \( \angle LMK \) (45°) is larger than \( \angle KMN \) (30°) and both angles add up to \( 75^\circ \) when combined with \( \angle KLM \) (unknown), we can conclude the order of the rays based on their angles.
The ray arrangement is usually as follows when looking from the vertex of the larger angle:
- Ray \( KL \) creates an angle of 45° with ray \( KM \) to the right (50% of the angle).
- Ray \( KM \) creates a 30° angle with ray \( KL \) to the left.
Thus, the ray \( KG \) from point K that corresponds to the angle 45° is situated in between the other two rays \( KL \) and \( KM \).
Therefore, the ray \( KL \) is between rays \( KM \) and \( ML \).
Thus, the answer is that ray KL is between rays KM and LM.