Question

given ab || dc
∠a ≅ ∠d
prove △abc ≅ △dcb
isabelle proves that the triangles are congruent by using the parallel lines to determine a second set of angles are congruent. what statement and reason could she have used?
∠abc ≅ ∠bac, corresponding angles of parallel lines are congruent
∠cab ≅ ∠dcb, alternate interior angles of parallel lines are congruent
∠abc ≅ ∠dcb, alternate interior angles of parallel lines are congruent
∠acd ≅ ∠abd, corresponding angles of parallel lines are congruent

Explanation:

Step1: Recall properties of parallel lines

When two parallel lines (\(AB\parallel DC\)) are cut by a transversal (\(BC\) in this case for angles \( \angle ABC\) and \( \angle DCB\)), alternate - interior angles are congruent.

Step2: Analyze each option

• Option A: \( \angle ABC\) and \( \angle BAC\) are not formed by parallel lines \(AB\) and \(DC\) and a transversal.
• Option B: \( \angle CAB\) and \( \angle DCB\) are not alternate - interior angles for the parallel lines \(AB\parallel DC\).
• Option C: Since \(AB\parallel DC\) and \(BC\) is a transversal, by the alternate - interior angles theorem, \( \angle ABC\cong\angle DCB\).
• Option D: \( \angle ACD\) and \( \angle ABD\) are not formed by the parallel lines \(AB\parallel DC\) and a single transversal in a way that would make them corresponding or alternate - interior angles.

Answer:

C. \( \angle ABC\cong\angle DCB \), alternate interior angles of parallel lines are congruent