Question

ik || qs and tv || ik. complete the proof that ∠put ≅ ∠srw. statement reason 1 ik || qs given 2 tv || ik given 3 ∠put ≅ ∠ijp 4 ∠ijp ≅ ∠srw alternate interior angles theorem 5 ∠put ≅ ∠srw transitive property of congruence

Explanation:

Step1: Identify parallel - line relationships

Given $\overleftrightarrow{IK}\parallel\overleftrightarrow{QS}$ and $\overleftrightarrow{TV}\parallel\overleftrightarrow{IK}$.

Step2: Use alternate - interior angles theorem for $\angle PUT$ and $\angle IJP$

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

Step3: Use alternate - interior angles theorem for $\angle IJP$ and $\angle SRW$

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

Step4: Apply transitive property

By the Transitive Property of Congruence, if $\angle PUT\cong\angle IJP$ and $\angle IJP\cong\angle SRW$, then $\angle PUT\cong\angle SRW$.

Answer:

The proof is completed as shown above. The reason for $\angle PUT\cong\angle IJP$ is the Alternate Interior Angles Theorem (because $\overleftrightarrow{TV}\parallel\overleftrightarrow{IK}$).