Question: RT \parallel GI and UW \parallel GI. Complete the proof that $m\angle RSX + m\angle QVU = 180^\circ$.\n\n1 \leftrightarrow RT \parallel \leftrightarrow GI Given\n2 \leftrightarrow UW \parallel \leftrightarrow GI Given\n3 \angle RSX \cong \angle GHX Corresponding Angles Theorem\n4 m\angle GHX + m\angle QVU = 180^\circ Linear Pair Postulate\n5 m\angle RSX + m\angle QVU = 180^\circ Substitution
1. **State the problem:** We are given that $\overleftrightarrow{RT} \parallel \overleftrightarrow{GI}$ and $\overleftrightarrow{UW} \parallel \overleftrightarrow{GI}$. We need to prove that $m\angle RSX + m\angle QVU = 180^\circ$.\n\n2. **Given:**\n- $\overleftrightarrow{RT} \parallel \overleftrightarrow{GI}$\n- $\overleftrightarrow{UW} \parallel \overleftrightarrow{GI}$\n\n3. **Use the Corresponding Angles Theorem:**\nSince $\overleftrightarrow{RT} \parallel \overleftrightarrow{GI}$ and the transversal passes through points $R, S, X, H, G, I$, the corresponding angles $\angle RSX$ and $\angle GHX$ are congruent.\n\n$$\angle RSX \cong \angle GHX$$\n\n4. **Apply the Linear Pair Postulate:**\nAngles $\angle GHX$ and $\angle QVU$ form a linear pair, so their measures add up to $180^\circ$.\n\n$$m\angle GHX + m\angle QVU = 180^\circ$$\n\n5. **Substitute the congruent angle:**\nSince $\angle RSX \cong \angle GHX$, their measures are equal, so substitute $m\angle RSX$ for $m\angle GHX$ in the equation above:\n\n$$m\angle RSX + m\angle QVU = 180^\circ$$\n\n**Final answer:**\n$$m\angle RSX + m\angle QVU = 180^\circ$$\n\nThis completes the proof that the sum of the measures of angles $RSX$ and $QVU$ is $180^\circ$.