Question

rt || gi and uw || gi. complete the proof that ∠tsx ≅ ∠qvu.
statement reason
1 rt || gi given
2 uw || gi given
3 ∠qvu ≅ ∠ghq
4 ∠ghq ≅ ∠tsx alternate interior angles theorem
5 ∠tsx ≅ ∠qvu

Explanation:

Step1: Identify parallel - line relationships

Given $\overleftrightarrow{RT}\parallel\overleftrightarrow{GI}$ and $\overleftrightarrow{UW}\parallel\overleftrightarrow{GI}$.

Step2: Justify $\angle QVU\cong\angle GHQ$

Since $\overleftrightarrow{UW}\parallel\overleftrightarrow{GI}$, by the Alternate Interior Angles Theorem, when a transversal intersects two parallel lines, alternate - interior angles are congruent. So, $\angle QVU\cong\angle GHQ$.

Step3: Justify $\angle GHQ\cong\angle TSX$

Since $\overleftrightarrow{RT}\parallel\overleftrightarrow{GI}$, by the Alternate Interior Angles Theorem, $\angle GHQ\cong\angle TSX$.

Step4: Use the Transitive Property of Congruence

If $\angle QVU\cong\angle GHQ$ and $\angle GHQ\cong\angle TSX$, then by the Transitive Property of Congruence (if $a = b$ and $b = c$, then $a = c$ for congruent angles), $\angle TSX\cong\angle QVU$.

Answer:

1. Alternate Interior Angles Theorem; 5. Transitive Property of Congruence