Question

in the given figure, which lines are parallel? justify your answer. determine the pair(s) of parallel lines. select all that apply. a. ae||gn by the same - side interior angles theorem b. gn||en by the alternate exterior angles theorem c. ag||en by the corresponding angles theorem d. ae||ag by the alternate interior angles theorem e. ae||en by the alternate interior angles theorem f. gn||ag by the same - side interior angles theorem

Explanation:

Step1: Recall angle - related parallel - line theorems

Parallel - line theorems include corresponding angles, alternate interior angles, alternate exterior angles, and same - side interior angles theorems. Each theorem gives a condition for two lines to be parallel based on the equality or supplementary relationship of angles formed by a transversal.

Step2: Analyze each option

• A: If $\overleftrightarrow{AE}\parallel\overleftrightarrow{GN}$ by the Same - Side Interior Angles Theorem, then the sum of same - side interior angles formed by a transversal with $\overleftrightarrow{AE}$ and $\overleftrightarrow{GN}$ must be $180^{\circ}$.
• B: If $\overleftrightarrow{GN}\parallel\overleftrightarrow{EN}$ by the Alternate Exterior Angles Theorem, the alternate exterior angles formed by a transversal with $\overleftrightarrow{GN}$ and $\overleftrightarrow{EN}$ must be equal. But $\overleftrightarrow{GN}$ and $\overleftrightarrow{EN}$ share a common point $N$, so they cannot be parallel.
• C: If $\overleftrightarrow{AG}\parallel\overleftrightarrow{EN}$ by the Corresponding Angles Theorem, the corresponding angles formed by a transversal with $\overleftrightarrow{AG}$ and $\overleftrightarrow{EN}$ must be equal.
• D: If $\overleftrightarrow{AE}\parallel\overleftrightarrow{AG}$ by the Alternate Interior Angles Theorem, $\overleftrightarrow{AE}$ and $\overleftrightarrow{AG}$ share a common point $A$, so they cannot be parallel.
• E: If $\overleftrightarrow{AE}\parallel\overleftrightarrow{EN}$ by the Alternate Interior Angles Theorem, $\overleftrightarrow{AE}$ and $\overleftrightarrow{EN}$ share a common point $E$, so they cannot be parallel.
• F: If $\overleftrightarrow{GN}\parallel\overleftrightarrow{AG}$ by the Same - Side Interior Angles Theorem, $\overleftrightarrow{GN}$ and $\overleftrightarrow{AG}$ share a common point $G$, so they cannot be parallel.

Answer:

A. $\overleftrightarrow{AE}\parallel\overleftrightarrow{GN}$ by the Same - Side Interior Angles Theorem
C. $\overleftrightarrow{AG}\parallel\overleftrightarrow{EN}$ by the Corresponding Angles Theorem