Question

given $\angle 8 \cong \angle 12$, 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:

Step1: Identify Angle Relationship

∠8 and ∠12 are alternate exterior angles for lines \( c \) and \( d \) (with the transversal being the line intersecting them).

Step2: Apply Converse Theorem

The Converse of the Alternate Exterior Angles Theorem states that if alternate exterior angles are congruent, then the lines are parallel. Since ∠8 ≅ ∠12, by this converse theorem, \( c \parallel d \).

Answer:

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