1. **Problem Statement:** Given a circle with center O and points A, B, C, D, E on or inside the circle, with lines AB, BC, CD, DE, AE drawn as described, and angles $x$ at vertex A and $33^\circ$ at vertex E, find the value of angle $x$. 2. **Relevant Theorem:** The angle formed outside a circle by two secants (lines intersecting the circle) is half the difference of the measures of the intercepted arcs. 3. **Explanation:** - Angle $x$ at point A is formed by secants AB and AE. - The intercepted arcs are arc BE and arc CD (or the arcs subtended by these secants). - The angle outside the circle is given by: $$x = \frac{1}{2} |\text{arc } BE - \text{arc } DE|$$ 4. **Given:** - Angle at E is $33^\circ$, which is an inscribed angle. - Inscribed angle theorem states that an inscribed angle is half the measure of its intercepted arc. - So, arc BD (intercepted by angle at E) is: $$\text{arc } BD = 2 \times 33^\circ = 66^\circ$$ 5. **Using the circle properties:** - The total circle is $360^\circ$. - The arcs around the circle add up to $360^\circ$. - If arc BD is $66^\circ$, then the remaining arcs sum to $360^\circ - 66^\circ = 294^\circ$. 6. **Calculate angle $x$:** - Angle $x$ is half the difference of arcs BE and DE. - Since points B, C, D, E lie on the circle, and given the arcs, angle $x$ equals $33^\circ$ by the external angle theorem for secants. **Final answer:** $$x = 33^\circ$$