Question

tv || ik and qs || ik. complete the proof that ( mangle tuw + mangle prq = 180^circ ).
diagram: lines ( overleftrightarrow{tv} ) (t upward, v downward), ( overleftrightarrow{ik} ) (i upward, k downward), ( overleftrightarrow{qs} ) (q upward, s downward), and transversal ( overleftrightarrow{wp} ) (w left, p right) intersecting ( overleftrightarrow{tv} ) at u, ( overleftrightarrow{ik} ) at j, ( overleftrightarrow{qs} ) at r.

statement	reason
2. ( overleftrightarrow{qs} parallel overleftrightarrow{ik} )	given
3. ( angle tuw cong angle ijw )	corresponding angles theorem
4. ( mangle ijw + mangle prq = 180^circ )	blank
5. ( mangle tuw + mangle prq = 180^circ )	substitution

Explanation:

Step1: Identify the relationship between lines

Since \(\overleftrightarrow{TV} \parallel \overleftrightarrow{IK}\) and \(\overleftrightarrow{QS} \parallel \overleftrightarrow{IK}\), by the Transitive Property of Parallel Lines, \(\overleftrightarrow{TV} \parallel \overleftrightarrow{QS}\). Now, look at \(\angle IJW\) and \(\angle PRQ\).

Step2: Determine the angle relationship

\(\overleftrightarrow{WJPR}\) is a transversal cutting the parallel lines \(\overleftrightarrow{IK}\) (or \(\overleftrightarrow{TV}\) and \(\overleftrightarrow{QS}\)). \(\angle IJW\) and \(\angle PRQ\) are same - side interior angles. By the Same - Side Interior Angles Theorem, if two parallel lines are cut by a transversal, then the same - side interior angles are supplementary. So, \(m\angle IJW + m\angle PRQ=180^{\circ}\) because \(\overleftrightarrow{IK} \parallel \overleftrightarrow{QS}\) (given) and \(\overleftrightarrow{WJPR}\) is a transversal, so same - side interior angles are supplementary.

Answer:

Same - Side Interior Angles Theorem (or If two parallel lines are cut by a transversal, then same - side interior angles are supplementary)