Question

statement reason
1 $overleftrightarrow{rt}paralleloverleftrightarrow{jk}$ given
2 $overleftrightarrow{jk}paralleloverleftrightarrow{uw}$ given
3 $angle wvxcongangle kjx$
4 $mangle kjx + mangle qst=180^{circ}$
5 $mangle wvx + mangle qst = 180^{circ}$

Explanation:

Step1: Transitive property of parallel lines

If \(RT\parallel JK\) and \(JK\parallel UW\), then by the transitive - property of parallel lines, \(RT\parallel UW\).

Step2: Corresponding angles postulate

Since \(RT\parallel UW\), \(\angle WVX\cong\angle KJX\) because they are corresponding angles.

Step3: Same - side interior angles theorem

\(RT\parallel JK\), and \(\angle KJX\) and \(\angle QST\) are same - side interior angles. So, \(m\angle KJX + m\angle QST=180^{\circ}\) by the same - side interior angles theorem.

Step4: Substitution property

Since \(\angle WVX\cong\angle KJX\), we can substitute \(\angle WVX\) for \(\angle KJX\) in the equation \(m\angle KJX + m\angle QST = 180^{\circ}\), getting \(m\angle WVX + m\angle QST=180^{\circ}\).

Answer:

1. Corresponding angles postulate; 4. Same - side interior angles theorem; 5. Substitution property.