1. **Problem statement:** Find the unknown angle $x$ in the given figure with two tangent circles and given angles $37^\circ$, $60^\circ$, and $80^\circ$. 2. **Understanding the figure and angles:** - Points $T$ and $S$ lie on line $AB$. - Larger circle passes through $E, D, T$. - Smaller circle passes through $D, C, S$. - Given angles: $\angle T = 37^\circ$, $\angle$ near $D$ inside larger circle is $x^\circ$, $\angle$ between $D$ and $C$ inside smaller circle is $80^\circ$, and another angle near $D$ inside larger circle is $60^\circ$. 3. **Key circle theorem:** - The angle between a tangent and chord through the point of contact equals the angle in the alternate segment. 4. **Step-by-step solution:** - Since $T$ lies on line $AB$ and $\angle T = 37^\circ$, this is the angle between tangent $AB$ and chord $DT$ in the larger circle. - By the alternate segment theorem, $\angle EDT = 37^\circ$ (angle in the alternate segment to tangent at $T$). - Given $\angle EDC = 60^\circ$ (angle near $D$ inside larger circle), and $\angle DCS = 80^\circ$ (angle inside smaller circle). - The unknown angle $x$ is $\angle EDC$ adjacent to $60^\circ$ and $80^\circ$ angles. - Since $D$ lies on both circles, angles around $D$ sum to $180^\circ$ (straight line or triangle sum). - Sum of angles at $D$ in the larger circle: $x + 60^\circ + 37^\circ = 180^\circ$. - Calculate $x$: $$x = 180^\circ - 60^\circ - 37^\circ = 83^\circ$$ 5. **Final answer:** $$\boxed{83^\circ}$$