Question

given ∠7 ≅ ∠9, which lines, if any, must be parallel based on the given information? justify your conclusion. a. ( a parallel b ), converse of the alternate exterior angles theorem c. ( c parallel d ), converse of the same - side interior angles theorem a. ( a parallel b ), converse of the alternate interior angles theorem not enough information to make a conclusion

Explanation:

Step1: Identify Angle Positions

$\angle 7$ and $\angle 9$: $\angle 7$ is at the intersection of line $b$ and transversal, $\angle 9$ at line $a$ and same transversal. They are alternate interior angles (between lines $a,b$ and transversal).

Step2: Apply Converse Theorem

Converse of Alternate Interior Angles Theorem: If alternate interior angles are congruent, then the lines cut by the transversal are parallel. Since $\angle 7 \cong \angle 9$, lines $a$ and $b$ must be parallel (by Converse of Alternate Interior Angles Theorem).

Step3: Evaluate Options

• Option a: Alternate Exterior (not, $\angle 7,\angle 9$ are interior).
• Option b: Converse of Alternate Interior (matches, as $\angle 7\cong\angle 9$ (alternate interior) implies $a\parallel b$).
• Option c: Same - Side Interior (no, angles are congruent, not supplementary).
• Option d: Not enough (incorrect, as we can apply the theorem).

Answer:

$\boldsymbol{\text{a } \parallel \text{ b, Converse of the Alternate Interior Angles Theorem}}$ (corresponding to the option with this description)