Question

statement reason 1. $overleftrightarrow{ik}paralleloverleftrightarrow{su}$ given 2. $overleftrightarrow{ik}paralleloverleftrightarrow{vx}$ given 3. $angle xwycongangle kjy$ corresponding angles theorem 4. $mangle kjy + mangle rtu=180^{circ}$ 5. $mangle xwy + mangle rtu = 180^{circ}$ alternate exterior angles theorem alternate interior angles theorem converse of alternate exterior angles theorem converse of alternate interior angles theorem converse of corresponding angles theorem converse of same - side exterior angles theorem

Explanation:

Step1: Recall angle - relationship theorems

We know from the given parallel lines \(IK\parallel SU\) and \(IK\parallel VX\). We have \(\angle XWY\cong\angle KJY\) by the Corresponding Angles Theorem.

Step2: Analyze the relationship between \(\angle KJY\) and \(\angle RTU\)

Since \(IK\parallel SU\), and \(\angle KJY\) and \(\angle RTU\) are same - side interior angles. By the Same - Side Interior Angles Theorem, when two parallel lines are cut by a transversal, same - side interior angles are supplementary. So \(m\angle KJY + m\angle RTU=180^{\circ}\).

Step3: Substitute \(\angle KJY\) with \(\angle XWY\)

Because \(\angle XWY\cong\angle KJY\) (from step 1), we can substitute \(\angle KJY\) in the equation \(m\angle KJY + m\angle RTU = 180^{\circ}\) to get \(m\angle XWY + m\angle RTU=180^{\circ}\) by the Substitution Property of Equality.

Answer:

Same - Side Interior Angles Theorem