1. **Problem statement:** Find the unknown angle $b$ in the kite-shaped quadrilateral where one angle is $110^\circ$ and an adjacent angle is $20^\circ$. 2. **Step 1:** Recall that in a kite, two pairs of adjacent sides are equal, and one diagonal bisects the other. 3. **Step 2:** The sum of angles around a point is $360^\circ$. The kite has four angles, and the sum of interior angles in any quadrilateral is $360^\circ$. 4. **Step 3:** Given the top left angle is $110^\circ$ and the bottom adjacent angle is $20^\circ$, the diagonal divides the kite into two triangles. 5. **Step 4:** In the triangle containing angle $b$, the sum of angles is $180^\circ$. The known angles are $110^\circ$ and $20^\circ$. 6. **Step 5:** Calculate $b$ using the triangle angle sum property: $$b = 180^\circ - 110^\circ - 20^\circ = 50^\circ$$ 7. **Answer:** The unknown angle $b$ is $50^\circ$.