1. **Problem statement:** Given \(\angle AYR = 30^\circ\) and \(\angle AYP = 45^\circ\), find \(\angle RYP\). 2. **Understanding the problem:** Points \(A, Y, R, P\) lie on a circle with center \(X\). The angles \(\angle AYR\) and \(\angle AYP\) are inscribed angles subtended by arcs on the circle. 3. **Key property:** In a circle, the measure of an inscribed angle is half the measure of the intercepted arc. 4. **Step-by-step solution:** - Let the arcs intercepted by these angles be \(\overset{\frown}{AR}\) and \(\overset{\frown}{AP}\). - Since \(\angle AYR = 30^\circ\), the arc \(\overset{\frown}{AR} = 2 \times 30^\circ = 60^\circ\). - Since \(\angle AYP = 45^\circ\), the arc \(\overset{\frown}{AP} = 2 \times 45^\circ = 90^\circ\). - The arc \(\overset{\frown}{RP}\) is the difference between arcs \(\overset{\frown}{AP}\) and \(\overset{\frown}{AR}\): $$\overset{\frown}{RP} = 90^\circ - 60^\circ = 30^\circ$$ - The angle \(\angle RYP\) intercepts arc \(\overset{\frown}{RP}\), so: $$\angle RYP = \frac{1}{2} \times \overset{\frown}{RP} = \frac{1}{2} \times 30^\circ = 15^\circ$$ 5. **Final answer:** $$\boxed{15^\circ}$$ This means \(\angle RYP = 15^\circ\).