Question

complete the proof that ∠tuz≅∠sxy.
statement\treason
1 (overleftrightarrow{tv}paralleloverleftrightarrow{gi})\tgiven
2 (overleftrightarrow{wy}paralleloverleftrightarrow{gi})\tgiven
3 ∠sxy≅∠ihs\tcorresponding angles theorem
4 ∠ihs≅∠tuz\t
5 ∠tuz≅∠sxy\t

Explanation:

Step1: Identify parallel - line properties

Since $\overleftrightarrow{TV}\parallel\overleftrightarrow{GI}$ and $\overleftrightarrow{WY}\parallel\overleftrightarrow{GI}$, we use angle - relation theorems.

Step2: Recall vertical - angles property

$\angle IHS$ and $\angle TUZ$ are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So the reason for $\angle IHS\cong\angle TUZ$ is the Vertical Angles Theorem.

Step3: Use transitive property of congruence

We know $\angle SXY\cong\angle IHS$ and $\angle IHS\cong\angle TUZ$. By the Transitive Property of Congruence (if $\angle A\cong\angle B$ and $\angle B\cong\angle C$, then $\angle A\cong\angle C$), we can conclude $\angle TUZ\cong\angle SXY$. The reason for $\angle TUZ\cong\angle SXY$ is the Transitive Property of Congruence.

Answer:

1. Vertical Angles Theorem
2. Transitive Property of Congruence