Question

statement reason 1 $overleftrightarrow{gi}paralleloverleftrightarrow{rt}$ given 2 $overleftrightarrow{uw}paralleloverleftrightarrow{gi}$ given 3 $angle qvwcongangle ihq$ corresponding angles theorem 4 $mangle ihq + mangle tsx=180^{circ}$ 5 $mangle tsx + mangle qvw = 180^{circ}$

Explanation:

Step1: Identify parallel - line relationships

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

Step2: Apply corresponding - angles theorem

Since $\overleftrightarrow{UW}\parallel\overleftrightarrow{GI}$, by the Corresponding Angles Theorem, $\angle QVW\cong\angle IHQ$.

Step3: Use same - side interior angles property

Because $\overleftrightarrow{GI}\parallel\overleftrightarrow{RT}$, $\angle IHQ$ and $\angle TSX$ are same - side interior angles. So, $m\angle IHQ + m\angle TSX=180^{\circ}$ (Same - Side Interior Angles Postulate).

Step4: Substitute equal angles

Since $\angle QVW\cong\angle IHQ$, we can substitute $\angle IHQ$ with $\angle QVW$ in the equation $m\angle IHQ + m\angle TSX = 180^{\circ}$. So, $m\angle TSX + m\angle QVW=180^{\circ}$.

Answer:

1. Reason: Same - Side Interior Angles Postulate
2. Reason: Substitution Property of Equality