Question

given \\(\overline{bc} \parallel \overline{ad}\\), complete the flowchart proof below.
(there is a diagram of a quadrilateral (abcd) with (ac) as a diagonal, angles at (b) and (d) marked with congruence arcs, and a flowchart proof with boxes: (overline{bc} parallel overline{ad}) (reason: given), (angle bca cong angle dac) (reason: select reason), (angle b cong angle d) (reason: select reason), (overline{ac} cong overline{ac}) (reason: select reason), and (\triangle abc cong \triangle cda) (reason: select reason))

Explanation:

1. For $\angle BCA \cong \angle DAC$: Since $\overline{BC} \parallel \overline{AD}$, these are alternate interior angles formed by transversal $\overline{AC}$, so they are congruent.
2. For $\overline{AC} \cong \overline{AC}$: This is a side that is shared by both $\triangle ABC$ and $\triangle CDA$, so it is congruent to itself by the reflexive property.
3. For $\triangle ABC \cong \triangle CDA$: We have $\angle B \cong \angle D$, $\angle BCA \cong \angle DAC$, and $\overline{AC} \cong \overline{AC}$, which fits the AAS (Angle-Angle-Side) congruence criterion.

Answer:

• $\angle BCA \cong \angle DAC$ Reason: Alternate Interior Angles Theorem
• $\overline{AC} \cong \overline{AC}$ Reason: Reflexive Property of Congruence
• $\triangle ABC \cong \triangle CDA$ Reason: AAS (Angle-Angle-Side) Congruence Postulate