Question

in the given figure, which lines are parallel? justify your answer. determine the pairs of parallel lines. choose the correct answer below. a. na||cg by the converse of the alternate interior angles theorem b. na||cg by the converse of the corresponding angles theorem c. nc||ag by the converse of the alternate interior angles theorem d. na||cg by the converse of the same - side interior angles postulate e. nc||ag by the converse of the vertical angles theorem

Explanation:

Step1: Recall parallel - line theorems

Parallel - line theorems help us determine if lines are parallel. The Converse of the Alternate Interior Angles Theorem states that if alternate interior angles are congruent, then the lines are parallel. The Converse of the Corresponding Angles Theorem states that if corresponding angles are congruent, then the lines are parallel. The Converse of the Same - Side Interior Angles Postulate states that if same - side interior angles are supplementary, then the lines are parallel. The Vertical Angles Theorem is about vertical angles being congruent and is not used to prove parallel lines.

Step2: Analyze the given figure and options

In the given figure, we need to identify the pairs of parallel lines based on the correct theorem. Looking at the options:

• Option A: $\overleftrightarrow{NA}\parallel\overleftrightarrow{CG}$ by the Converse of the Alternate Interior Angles Theorem. This is a valid way to prove two lines are parallel if the alternate interior angles formed by a transversal are congruent.
• Option B: There is no indication from the problem - statement that we are using corresponding angles to prove $\overleftrightarrow{NA}\parallel\overleftrightarrow{CG}$.
• Option C: There is no information suggesting that $\overleftrightarrow{NC}\parallel\overleftrightarrow{AG}$ based on alternate interior angles.
• Option D: There is no information suggesting that $\overleftrightarrow{NA}\parallel\overleftrightarrow{CG}$ based on same - side interior angles.
• Option E: The Converse of the Vertical Angles Theorem is not a valid way to prove parallel lines.

Answer:

A. $\overleftrightarrow{NA}\parallel\overleftrightarrow{CG}$ by the Converse of the Alternate Interior Angles Theorem