1. **State the problem:** Find the values of angles a, d, and e in the circle with center O, points L, M, N, R on the circumference, tangent PQ at R, given $\angle PRL = 48^\circ$ and $\angle RON = 156^\circ$. 2. **Recall key theorems and facts:** - The angle at the center $\angle RON$ is twice the angle at the circumference subtended by the same arc: $\angle RON = 2 \times \angle RLN$. - The tangent to a circle is perpendicular to the radius at the point of contact: $\angle ORP = 90^\circ$. - Angles in a triangle sum to $180^\circ$. 3. **Find angle d:** Since $\angle RON = 156^\circ$, the angle at the circumference subtended by arc RN is $$\angle RLN = \frac{\angle RON}{2} = \frac{156^\circ}{2} = 78^\circ.$$ So, $d = 78^\circ$. 4. **Find angle e:** At point R, $\angle PRL = 48^\circ$ is given. Since PQ is tangent at R and OR is radius, $\angle ORP = 90^\circ$. Triangle PRL has angles $\angle PRL = 48^\circ$, $\angle RLP = e$, and $\angle LPR$. But $e$ is the angle between RL and tangent PQ, so $e = 90^\circ - 48^\circ = 42^\circ$. 5. **Find angle a:** Point M lies on the circumference; angle a is subtended by arc RL. Since $d = 78^\circ$ subtends arc RN, and $\angle RON = 156^\circ$, the remaining arc RL is $360^\circ - 156^\circ = 204^\circ$. Angle at circumference $a = \frac{204^\circ}{2} = 102^\circ$. **Final answers:** $$a = 102^\circ, \quad d = 78^\circ, \quad e = 42^\circ.$$