Question

Match the reasons with the statements in the proof to prove AB || DC given that AD is parallel to BC and AD = CB.

A) Ad || BC; AD = CB (CPCTE - Corresponding parts of congruent triangles are equal)
B) AC = AC (Given)
C) ∠2 = ∠3 (If alternate interior angles are congruent, then lines are parallel)
D) △ ACD = △ CAB (SAS - Side-Angle-Side)
E) ∠1 = ∠4 (If lines are parallel, then alternate interior angles are equal)
F) AB || DC (Reflexive property of equality)

Answer 1

The correct matching of statements to reasons in the proof are: ∠1 = ∠4 to establish ∠A = ∠D, △ ACD = △ CAB (SAS) to use the common side AC, and ∠2 = ∠3 to conclude AB || DC.

Step-by-step explanation:

The task is to match statements to reasons in a geometric proof that AB || DC, given that AD is parallel to BC and AD equals CB. Here's the correct matching:

1. (E) ∠1 = ∠4 (If lines are parallel, then alternate interior angles are equal) Which proves that ∠A = ∠D since they are alternate interior angles.
2. (D) △ ACD = △ CAB (SAS - Side-Angle-Side) because we know AD = CB (given), ∠A = ∠D (from previous step), and AC is common to both triangles.
3. (C) ∠2 = ∠3 (If alternate interior angles are congruent, then lines are parallel), thus establishing that AB || DC due to the congruence of alternate interior angles.
4. (F) AB || DC (Reflexive property of equality) directly states the conclusion of the proof based on the given theorem related to alternate interior angles. This is the final statement that concludes AB || DC.