1. **Problem 1:** Calculate the size of angle $x$ in the cyclic quadrilateral $ABEC$ where angle $D$ is $136^\circ$. 2. **Key fact:** Opposite angles in a cyclic quadrilateral sum to $180^\circ$. 3. Since $ABEC$ is cyclic, angles $x$ (at $A$) and $136^\circ$ (at $D$) are opposite angles. 4. Use the formula: $$x + 136^\circ = 180^\circ$$ 5. Solve for $x$: $$x = 180^\circ - 136^\circ = 44^\circ$$ 6. **Reason:** Opposite angles of a cyclic quadrilateral are supplementary. 7. **Problem 2:** Work out the size of angle $\theta$ at point $C$ formed by tangent $BC$ and line $DE$. 8. Given angles: $\angle A = 52^\circ$, $\angle D = 27^\circ$. 9. **Key fact:** The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment. 10. Tangent $BC$ touches the circle at $B$, so $\theta$ equals the angle in the alternate segment at $A$, which is $52^\circ$. 11. **Reason:** Alternate segment theorem. **Final answers:** $$x = 44^\circ$$ $$\theta = 52^\circ$$