Question

given $\angle 1 \cong \angle 7$, which lines, if any, must be parallel based on the given information? justify your conclusion.

\bigcirc $a \parallel b$, converse of the corresponding angles theorem

\bigcirc $c \parallel d$, converse of the alternate exterior angles theorem

\bigcirc $a \parallel b$, converse of the same - side interior angles theorem

\bigcirc not enough information to make a conclusion

Explanation:

1. First, identify the angles: \( \angle 1 \) and \( \angle 7 \) are alternate exterior angles.
2. The Converse of the Alternate Exterior Angles Theorem states that if two lines are cut by a transversal and the alternate exterior angles are congruent, then the lines are parallel.
3. Here, the transversal is the line between \( c \) and \( d \), and the lines cut by it are \( c \) and \( d \). Since \( \angle 1 \cong \angle 7 \) (alternate exterior angles), by the Converse of the Alternate Exterior Angles Theorem, \( c \parallel d \).

For the other options:

• Option A: \( \angle 1 \) and \( \angle 7 \) are not corresponding angles for lines \( a \) and \( b \), so the Converse of Corresponding Angles Theorem does not apply.
• Option C: \( \angle 1 \) and \( \angle 7 \) are not same - side interior angles for lines \( a \) and \( b \), so the Converse of Same - Side Interior Angles Theorem does not apply.
• Option D: There is enough information (using the Converse of Alternate Exterior Angles Theorem) to conclude \( c \parallel d \).

Answer:

B. \( c \parallel d \), Converse of the Alternate Exterior Angles Theorem