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 - angle relationship

$\angle{SUT}$ and $\angle{VUX}$ are vertical angles, so $\angle{SUT}\cong\angle{VUX}$ (Vertical angles are congruent).

Step2: Use transitive property

Since $\angle{UXW}\cong\angle{VUX}$ (given) and $\angle{SUT}\cong\angle{VUX}$, by the transitive property of congruence, $\angle{UXW}\cong\angle{SUT}$.

Step3: Apply converse of corresponding - angles theorem

$\angle{UXW}$ and $\angle{SUT}$ are corresponding angles. By the converse of the corresponding - angles theorem, if corresponding angles are congruent, then the lines are parallel. So $\overleftrightarrow{TV}\parallel\overleftrightarrow{WY}$.

Answer:

1. Reason: Vertical angles are congruent.
2. Reason: Transitive property of congruence.