Question

given the following information, determine which lines, if any, are parallel. state the theorem that justifies your answer.
∠4 ≅ ∠9
a) a || b; consecutive exterior angles converse
b) c || d; consecutive interior angles converse
c) a || b; alternate interior angles converse
d) c || d; alternate exterior angles converse

Explanation:

Step1: Identify Angle Positions

$\angle 4$ and $\angle 9$: $\angle 4$ is an exterior angle relative to the transversal cutting lines \(c\) and \(d\), and $\angle 9$ is on the other side of the transversal. They are alternate exterior angles.

Step2: Apply Converse Theorem

The Alternate Exterior Angles Converse states that if two lines are cut by a transversal and the alternate exterior angles are congruent, then the lines are parallel. Since $\angle 4 \cong \angle 9$, by this converse, lines \(c\) and \(d\) are parallel.

Step3: Evaluate Options

• Option A: Incorrect lines and theorem.
• Option B: Incorrect theorem (consecutive interior, not alternate exterior).
• Option C: Incorrect lines and theorem.
• Option D: Matches \(c \parallel d\) and Alternate Exterior Angles Converse.

Answer:

D. \(c \parallel d\); Alternate Exterior Angles Converse