Question

∠tsv≅∠svu. complete the proof that uw⃡∥rt⃡. statement 1. ∠tsv≅∠svu 2. ∠qsr = ∠tsv 3. ∠svu = ∠qsr 4. uw⃡∥rt⃡ reason given *

Explanation:

Step1: Given information

$\angle TSV\cong\angle SVU$ (Given)

Step2: Transitive - property of congruence

Since $\angle QSR=\angle TSV$ (Vertical - angles are congruent) and $\angle SVU=\angle QSR$ (Given), by the transitive property of congruence, we can establish a relationship between the angles formed by the lines.

Step3: Alternate - interior angles

$\angle TSV$ and $\angle SVU$ are alternate - interior angles. If alternate - interior angles are congruent, then the lines are parallel.
If $\angle TSV\cong\angle SVU$, then $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$ (If alternate - interior angles are congruent, then the lines are parallel)

Answer:

1. $\angle TSV\cong\angle SVU$; Given
2. $\angle QSR = \angle TSV$; Vertical angles are congruent
3. $\angle SVU=\angle QSR$; Given (or Transitive property using 1 and 2)
4. $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$; If alternate - interior angles are congruent, then the lines are parallel