Question

complete the proof that ∠put≅∠srw.
statement reason

1. (overleftrightarrow{gi}paralleloverleftrightarrow{tv}) given
2. (overleftrightarrow{qs}paralleloverleftrightarrow{gi}) given
3. ∠put≅∠ghp
4. ∠ghp≅∠srw alternate interior angles theorem
5. ∠put≅∠srw

Explanation:

Step1: Identify parallel - line relationships

We are given that $\overleftrightarrow{GI}\parallel\overleftrightarrow{TV}$ and $\overleftrightarrow{QS}\parallel\overleftrightarrow{GI}$.

Step2: Apply alternate - interior angles theorem for $\overleftrightarrow{GI}$ and $\overleftrightarrow{TV}$

Since $\overleftrightarrow{GI}\parallel\overleftrightarrow{TV}$, by the Alternate Interior Angles Theorem, $\angle PUT\cong\angle GHP$.

Step3: Apply alternate - interior angles theorem for $\overleftrightarrow{QS}$ and $\overleftrightarrow{GI}$

Since $\overleftrightarrow{QS}\parallel\overleftrightarrow{GI}$, by the Alternate Interior Angles Theorem, $\angle GHP\cong\angle SRW$.

Step4: Use the transitive property of congruence

If $\angle PUT\cong\angle GHP$ and $\angle GHP\cong\angle SRW$, then by the transitive property of congruence, $\angle PUT\cong\angle SRW$.

Answer:

The reason for $\angle PUT\cong\angle GHP$ is the Alternate Interior Angles Theorem (because $\overleftrightarrow{GI}\parallel\overleftrightarrow{TV}$), and the reason for the last step (Statement 5) is the Transitive Property of Congruence (since $\angle PUT\cong\angle GHP$ and $\angle GHP\cong\angle SRW$).