Question

eg || tv and eg || wy. ∠sut ≅ ∠efs. prove that tv || wy.

Explanation:

Step1: Identify the reason for \( \angle EFS\cong\angle SXW \)

Since \( \overrightarrow{EG}\parallel\overrightarrow{WY} \), by the Alternate Interior Angles Theorem, \( \angle EFS\cong\angle SXW \).

Step2: Identify the reason for \( \angle SXW\cong\angle SUT \)

Given \( \angle EFS\cong\angle SUT \) and from Step 1 \( \angle EFS\cong\angle SXW \), by the Transitive Property of Congruence (if \( a = b \) and \( b = c \), then \( a = c \)), \( \angle SXW\cong\angle SUT \).

Step3: Identify the reason for \( \overrightarrow{TV}\parallel\overrightarrow{WY} \)

Since \( \overrightarrow{EG}\parallel\overrightarrow{TV} \) (given) and we have shown \( \angle SXW\cong\angle SUT \), and \( \angle SXW \) and \( \angle SUT \) are corresponding angles. By the Corresponding Angles Converse (if corresponding angles are congruent, then the lines are parallel), \( \overrightarrow{TV}\parallel\overrightarrow{WY} \).

Answer:

For \( \angle EFS\cong\angle SXW \): Alternate Interior Angles Theorem.
For \( \angle SXW\cong\angle SUT \): Transitive Property of Congruence.
For \( \overrightarrow{TV}\parallel\overrightarrow{WY} \): Corresponding Angles Converse.