Question

statement reason 1 $overleftrightarrow{rt}paralleloverleftrightarrow{hj}$ given 2 $overleftrightarrow{hj}paralleloverleftrightarrow{uw}$ given 3 $angle wvxcongangle jix$ corresponding angles theorem 4 $angle jixcongangle qsr$ 5 $angle qsrcongangle wvx$ transitive property of congruence

Explanation:

Step1: Identify parallel - line relationships

We are given $\overleftrightarrow{RT}\parallel\overleftrightarrow{HJ}$ and $\overleftrightarrow{HJ}\parallel\overleftrightarrow{UW}$ from statements 1 and 2.

Step2: Apply corresponding - angles theorem

From $\overleftrightarrow{UW}\parallel\overleftrightarrow{HJ}$, by the Corresponding Angles Theorem, $\angle WVX\cong\angle JIX$ (statement 3).

Step3: Find another pair of congruent angles

We need to find the reason for $\angle JIX\cong\angle QSR$. Since $\overleftrightarrow{RT}\parallel\overleftrightarrow{HJ}$, by the Corresponding Angles Theorem, $\angle JIX$ and $\angle QSR$ are corresponding angles, so $\angle JIX\cong\angle QSR$.

Step4: Use transitive property

By the Transitive Property of Congruence (statement 5), since $\angle WVX\cong\angle JIX$ and $\angle JIX\cong\angle QSR$, we can conclude $\angle QSR\cong\angle WVX$.

Answer:

The reason for $\angle JIX\cong\angle QSR$ is the Corresponding Angles Theorem because $\overleftrightarrow{RT}\parallel\overleftrightarrow{HJ}$.