Question

applications of probability and geometry
lines \overleftrightarrow{de} and \overleftrightarrow{ab} are parallel.
which angles are congruent?
angles a and c
angles a and e
angles b and c
angles c, d, and e

Explanation:

Step1: Recall Parallel Line Angles

When two parallel lines are cut by a transversal, alternate interior angles are congruent. Here, \( \overleftrightarrow{DE} \parallel \overleftrightarrow{AB} \), and transversals are \( \overline{CB} \) and \( \overline{CA} \).

Step2: Analyze Angle Pairs

• For angles \( A \) and \( E \): \( \overline{CA} \) is a transversal cutting \( \overleftrightarrow{DE} \) and \( \overleftrightarrow{AB} \). \( \angle A \) (on \( AB \)) and \( \angle E \) (on \( DE \)) are alternate interior angles, so they should be congruent.
• Check other options: Angles \( A \) and \( C \) don't have a parallel - line - transversal relation for congruence. Angles \( B \) and \( C \) are not alternate interior angles. Angles \( C \), \( D \), \( E \): \( \angle D \) and \( \angle E \) are on a straight line with \( \angle C \), but not all three are congruent.

Answer:

angles A and E