Perpendicular Lines 814Cc6

1. **State the problem:** We have two perpendicular planes $X$ and $Y$. Points $A$, $E$, $F$, and $G$ lie only in plane $X$. Points $R$ and $S$ lie in both planes $X$ and $Y$. Lines $EA$ and $FG$ are parallel. 2. **Understand the geometry:** Since $X$ and $Y$ are perpendicular planes, any line lying entirely in plane $X$ is perpendicular to any line lying entirely in plane $Y$ if they intersect. 3. **Analyze the lines:** - $RS$ lies in both planes $X$ and $Y$, so it is the line of intersection of the two planes. - Lines $EA$ and $FG$ lie in plane $X$ and are parallel. - Points $E$, $F$, $G$, and $A$ are in plane $X$ only. 4. **Check each line option for perpendicularity to $RS$:** - $ER$: connects points $E$ and $R$. Since $E$ is in $X$ only and $R$ is in both $X$ and $Y$, $ER$ lies in plane $X$ (because $E$ is only in $X$) and passes through $R$. - $FS$: connects $F$ and $S$. $F$ is in $X$ only, $S$ is in both $X$ and $Y$, so $FS$ lies in plane $X$. - $EF$: connects $E$ and $F$, both in $X$ only, so $EF$ lies entirely in plane $X$. 5. **Which lines can be perpendicular to $RS$?** Since $RS$ is the intersection line of planes $X$ and $Y$, any line perpendicular to $RS$ must lie in either plane $X$ or plane $Y$ and be perpendicular to $RS$. 6. **Lines $ER$ and $FS$ both connect a point in $X$ only to a point in both $X$ and $Y$, so they lie in plane $X$ and pass through $R$ or $S$. These lines can be perpendicular to $RS$ because they lie in plane $X$ and intersect $RS$ at $R$ or $S$. 7. **Line $EF$ lies entirely in plane $X$ but does not necessarily intersect $RS$ at $R$ or $S$, so it cannot be perpendicular to $RS$ at those points. **Final answer:** The two lines that could be perpendicular to $RS$ are $ER$ and $FS$.