Question: Top-left graph: two parallel horizontal lines AB and DE are cut by a slanted transversal CF; the upper interior angle is labeled $y$, the lower interior angle is labeled $x$, and the lower-left angle at the bottom intersection is $83^\circ$. (9) F R A 100^\circ S P 2x-20^\circ Q E Bottom-left graph: two parallel slanted lines are cut by a near-vertical transversal S; the upper intersection has angles $85^\circ$ and $a$, and the lower intersection has angles $d$, $147^\circ$, $c$, and $e$.

1. **Problem Statement:** We analyze the top-left graph where two parallel horizontal lines $AB$ and $DE$ are cut by a transversal $CF$. The angles given are: - Upper interior angle labeled $y$ - Lower interior angle labeled $x$ - Lower-left angle at the bottom intersection is $83^\circ$ 2. **Relevant Geometry Rules:** - Alternate interior angles between parallel lines are equal. - Corresponding angles between parallel lines are equal. - The sum of angles on a straight line is $180^\circ$. 3. **Step 1: Identify relationships between angles** Since $AB \parallel DE$ and $CF$ is a transversal, the upper interior angle $y$ and the lower interior angle $x$ are consecutive interior angles. 4. **Step 2: Use consecutive interior angles theorem** Consecutive interior angles are supplementary, so: $$x + y = 180^\circ$$ 5. **Step 3: Use given angles to find $x$ and $y$** The lower-left angle at the bottom intersection is $83^\circ$. This angle is adjacent to angle $x$ and forms a linear pair, so: $$x + 83^\circ = 180^\circ$$ 6. **Step 4: Solve for $x$** $$x = 180^\circ - 83^\circ = 97^\circ$$ 7. **Step 5: Solve for $y$ using $x + y = 180^\circ$** $$y = 180^\circ - x = 180^\circ - 97^\circ = 83^\circ$$ 8. **Final answers:** $$x = 97^\circ, \quad y = 83^\circ$$ This completes the solution for the first problem described.