Question: In the diagram of circle A, what is m\angle LMN? 75\degree 90\degree 120\degree 135\degree 270\degree Circle A is shown with center A; points L and N are on the circle and point M is outside the circle. Segments ML and MN are drawn as tangents from M to the circle, and the major arc from L to N is labeled 270\degree.
1. **Problem Statement:** We need to find the measure of angle $\angle LMN$ where $M$ is a point outside circle $A$, and $ML$ and $MN$ are tangents to the circle touching points $L$ and $N$ respectively. The major arc $LN$ measures $270^\circ$.
2. **Key Concept:** The angle formed between two tangents drawn from an external point to a circle is half the difference of the measures of the intercepted arcs.
3. **Formula:**
$$\angle LMN = \frac{1}{2} \times \text{(difference of arcs)}$$
4. **Identify arcs:**
- Major arc $LN = 270^\circ$
- Minor arc $LN = 360^\circ - 270^\circ = 90^\circ$
5. **Calculate angle:**
$$\angle LMN = \frac{1}{2} \times (270^\circ - 90^\circ)$$
$$\angle LMN = \frac{1}{2} \times 180^\circ$$
$$\angle LMN = 90^\circ$$
6. **Explanation:** The angle between the two tangents is half the difference of the major and minor arcs intercepted by those tangents.
**Final answer:** $\boxed{90^\circ}$