https://file.aiquickdraw.com/mathbot/file/1722741999792meeen1dd.jpg

To solve for \( x + y \) in the given diagram, let's analyze the angles step-by-step:

1. **Given Angles and Properties:**
   - \(\angle HNJ = 110^\circ\)
   - \(\angle HJL = 30^\circ\)
   - \(\angle KLM\) is the tangent to the circle at \( L \)
   - \( HNM \) is a straight line, implying that \(\angle HNL + \angle LNM = 180^\circ\)

2. **Inscribed Angle Theorem:**
   - Since \( HJ \) and \( JL \) are chords intersecting at \( J \), the angle \(\angle HJL\) subtended by arc \( HL \) is \( 30^\circ \).
   - The opposite angle subtended by the same arc \( HL \) is \(\angle HNL\). So, \(\angle HNL = \frac{1}{2} \times 30^\circ = 15^\circ\).

3. **Angle Sum around Point \( L \):**
   - Since \( KL \) is tangent to the circle at \( L \), \(\angle KLM = \angle HJL = 30^\circ\) (by the tangent-secant angle theorem).
   - \(\angle LNM = 110^\circ\) because \(\angle HNM\) is a straight line, thus \( \angle HNL + \angle LNM = 180^\circ\).

4. **Finding \( x \) and \( y \):**
   - Let \( \angle LNK = x \) and \( \angle NLM = y \).
   - From the angles at point \( N \): \( \angle HNL + \angle LNK = \angle HNJ \) so, \( 15^\circ + x = 110^\circ \rightarrow x = 95^\circ \).
   - From the angles at point \( L \): \( \angle NLM = y = 30^\circ\).

5. **Sum \( x + y \):**
   - Thus, \( x = 95^\circ \) and \( y = 30^\circ \).
   - So, \( x + y = 95^\circ + 30^\circ = 125^\circ \).

Therefore, \( x + y = 125^\circ \).

Would you like more details or have any questions about this solution?

Here are some related questions for further exploration:

1. What is the measure of \(\angle HNL\)?
2. Explain why \(\angle KLM = \angle HJL\).
3. How does the tangent-secant angle theorem apply here?
4. What is the measure of \(\angle JLM\)?
5. Find the measure of \(\angle KLN\).
6. How does the sum of angles around point \( N \) help in solving the problem?
7. Explain the inscribed angle theorem.
8. Why does \(\angle HNM\) being a straight line imply that \(\angle HNL + \angle LNM = 180^\circ\)?

**Tip:** When working with circle theorems, always consider the relationships between the angles subtended by arcs and those formed by tangents and secants.

Learn how to solve for x + y using circle geometry and angle relationships. This problem involves analyzing angles around points and on tangents, applying the Inscribed Angle Theorem and Tangent-Secant Angle Theorem. Suitable for high school students studying geometry.

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