1. **State the problem:** We have two lines defined by the equations: $$r: x + y = 2$$ $$s: y = (2 - k)x - 1, \quad k \in \mathbb{R}$$ We want to find the value(s) of $k$ for which the lines $r$ and $s$ are parallel. 2. **Rewrite line $r$ in slope-intercept form:** Starting with $$x + y = 2$$ subtract $x$ from both sides: $$y = 2 - x$$ or equivalently $$y = -1 \cdot x + 2$$ So the slope of line $r$ is $m_r = -1$. 3. **Identify the slope of line $s$:** Line $s$ is given as $$y = (2 - k)x - 1$$ The slope of $s$ is $$m_s = 2 - k$$ 4. **Condition for parallel lines:** Two lines are parallel if and only if their slopes are equal: $$m_r = m_s$$ Substitute the slopes: $$-1 = 2 - k$$ 5. **Solve for $k$:** Add $k$ to both sides and add $1$ to both sides: $$k - 1 = 2$$ $$k = 3$$ 6. **Answer:** The lines $r$ and $s$ are parallel if and only if $k = 3$. **Final answer:** (D) $k = 3$