Question

given (overline{ab} parallel overline{de}) leads to (angle bac cong angle edc); (overline{ab} cong overline{de}) leads to (\triangle abc cong \triangle dec); also (angle acb cong angle dce). options: vertical angles thm, aas, at interior angles thm, given

Explanation:

Step1: Analyze ∠BAC ≅ ∠EDC

Since \( \overline{AB} \parallel \overline{DE} \), \( \angle BAC \) and \( \angle EDC \) are alternate interior angles. By the Alternate Interior Angles Theorem, alternate interior angles formed by parallel lines and a transversal are congruent. So the reason for \( \angle BAC \cong \angle EDC \) is the Alternate Interior Angles Theorem.

Step2: Analyze ∠ACB ≅ ∠DCE

\( \angle ACB \) and \( \angle DCE \) are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So the reason for \( \angle ACB \cong \angle DCE \) is the Vertical Angles Theorem.

Step3: Analyze \( \triangle ABC \cong \triangle DEC \)

We have \( \angle BAC \cong \angle EDC \), \( \angle ACB \cong \angle DCE \), and \( \overline{AB} \cong \overline{DE} \) (given). This matches the AAS (Angle - Angle - Side) congruence criterion, which states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the triangles are congruent.

Answer:

• For \( \angle BAC \cong \angle EDC \): Alternate Interior Angles Theorem
• For \( \angle ACB \cong \angle DCE \): Vertical Angles Theorem
• For \( \triangle ABC \cong \triangle DEC \): AAS