Question

∠uxw≅∠vux. complete the proof that (overleftrightarrow{tv}paralleloverleftrightarrow{wy}).
statement\treason
1 ∠uxw≅∠vux\tgiven
2 ∠sut≅∠vux
3 ∠uxw≅∠sut
4 (overleftrightarrow{tv}paralleloverleftrightarrow{wy}\tconverse of corresponding angles theorem

Explanation:

Step1: Identify vertical - angles

$\angle SUT$ and $\angle VUX$ are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So, $\angle SUT\cong\angle VUX$.

Step2: Use the transitive property

Since $\angle U XW\cong\angle VUX$ (given) and $\angle SUT\cong\angle VUX$, by the transitive property of congruence, $\angle U XW\cong\angle SUT$.

Step3: Apply the converse of corresponding - angles theorem

$\angle U XW$ and $\angle SUT$ are corresponding angles. According to the Converse of the Corresponding Angles Theorem, if two corresponding angles are congruent, then the two lines are parallel. So, $\overleftrightarrow{TV}\parallel\overleftrightarrow{WY}$.

Answer:

The proof is completed as shown above, showing that $\overleftrightarrow{TV}\parallel\overleftrightarrow{WY}$ based on the given $\angle U XW\cong\angle VUX$ and geometric angle - relationship theorems.