1. **Problem statement:** We have a circle with points L, N, Q on the circumference, where LN = NQ, and RA is tangent at L. Given \(\angle MLN = 63^\circ\), we need to explain why \(\angle x = \angle NQL\) and find angles \(x\) and \(y\). 2. **Explanation why \(\angle x = \angle NQL\):** - Since LN = NQ, triangle LNQ is isosceles with \(\angle NQL = \angle NLQ\). - \(\angle MLN = 63^\circ\) is the angle between tangent RA and chord LN, so by the tangent-chord theorem, \(\angle MLN = \angle NQL = 63^\circ\). - Therefore, \(\angle x = \angle NQL = 63^\circ\). 3. **Find \(\angle x\):** - From above, \(\angle x = 63^\circ\). 4. **Find \(\angle y\):** - Triangle LNQ is isosceles with LN = NQ. - Sum of angles in triangle LNQ is \(180^\circ\). - Let \(\angle NLQ = \angle NQL = 63^\circ\). - Then \(\angle L N Q = 180^\circ - 2 \times 63^\circ = 54^\circ\). - \(\angle y = 54^\circ\). **Final answers:** - \(\angle x = 63^\circ\) - \(\angle y = 54^\circ\)