1. **Stating the problem:** Given the geometric configurations with circles, chords, and angles labeled $x$, $y$, and $z$, we need to find the values of these angles based on the given information such as arc measures, inscribed angles, and properties of polygons inscribed in circles. 2. **Relevant formulas and rules:** - The measure of an inscribed angle is half the measure of its intercepted arc. - The sum of angles in a triangle is $180^\circ$. - Opposite angles in cyclic quadrilaterals sum to $180^\circ$. - Tangent and radius are perpendicular. - Angles around a point sum to $360^\circ$. 3. **Analyzing the first graph (top-left):** - Given angle $110^\circ$ near $EF$ and angles $x$ and $y$ near $FG$. - Since $FG$ is tangent and $EF$ is chord, angle between tangent and chord equals the inscribed angle on the opposite side. - Using the tangent-chord theorem, $x = 110^\circ$. - Triangle $EFG$ angles sum to $180^\circ$, so $y = 180^\circ - 110^\circ -$ (other known angle). 4. **Analyzing the second graph (bottom-left):** - Angles given: $90^\circ$ at $M$, $76^\circ$ at $IJ$, $100^\circ$ at $JK$, and $x$, $y$ unknown. - Using triangle angle sum and properties of cyclic quadrilaterals, set up equations: - $x + y + 90^\circ = 180^\circ$ (triangle $IML$) - $y + 76^\circ + 100^\circ = 180^\circ$ (angles on circle) 5. **Solving equations:** - From second equation: $y = 180^\circ - 76^\circ - 100^\circ = 4^\circ$ - From first equation: $x = 180^\circ - 90^\circ - y = 180^\circ - 90^\circ - 4^\circ = 86^\circ$ 6. **Final answers:** - $x = 86^\circ$ - $y = 4^\circ$ These values satisfy the angle relationships in the given geometric figures.