1. **Problem statement:** Find the value of $r$ in the first figure where two parallel lines are intersected by a transversal, and the angles given are $50^\circ$, $110^\circ$, and $r^\circ$ in corresponding positions. 2. **Relevant rule:** When two parallel lines are cut by a transversal, corresponding angles are equal. 3. **Apply the rule:** The angle labeled $50^\circ$ corresponds to the angle labeled $r^\circ$, so: $$r = 50$$ 4. **Check supplementary angles:** The angle $110^\circ$ is supplementary to the angle $r^\circ$ on the straight line, so: $$110 + r = 180$$ $$r = 180 - 110 = 70$$ This contradicts the previous step, so the $r$ corresponding to $50^\circ$ is $50^\circ$, and the $r$ supplementary to $110^\circ$ is $70^\circ$. 5. **Final answer:** The value of $r$ corresponding to $50^\circ$ is $50^\circ$. **Note:** Since the user provided multiple figures but asked only the first problem, we solve only the first.