Question

1. given: \\(\overline{ab} \parallel \overline{cd}, \overline{bc} \parallel \overline{ad}\\) prove: \\(\overline{ad} \cong \overline{bc}\\) \\(\

$$\begin{array}{|c|c|} \\hline \\text{statement} & \\text{reason} \\\\ \\hline \\overline{ab} \\parallel \\overline{cd}, \\overline{bc} \\parallel \\overline{ad} & \\\\ \\hline \\angle abd \\cong \\angle bdc & \\text{aia} \\\\ \\hline & \\text{reflexive} \\\\ \\hline & \\\\ \\hline & \\\\ \\hline \\overline{ad} \\cong \\overline{bc} & \\\\ \\hline \\end{array}$$

\\)

Explanation:

Step1: Identify Given

The given is $\overline{AB} \parallel \overline{CD}, \overline{BC} \parallel \overline{AD}$, so the reason here is "Given".

Step2: Find Reflexive Angle

For the reflexive property, the statement should be $\overline{BD} \cong \overline{BD}$ (since a segment is congruent to itself).

Step3: Find Another AIA

Since $\overline{BC} \parallel \overline{AD}$, $\angle ADB \cong \angle CBD$ by AIA (Alternate Interior Angles).

Step4: Prove Triangles Congruent

Triangles $\triangle ABD$ and $\triangle CDB$ are congruent by ASA (Angle - Side - Angle: $\angle ABD \cong \angle BDC$, $\overline{BD} \cong \overline{BD}$, $\angle ADB \cong \angle CBD$).

Step5: Corresponding Parts Congruent

By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), $\overline{AD} \cong \overline{BC}$.

Filling the table:

Statement	Reason
$\angle ABD \cong \angle BDC$	AIA
$\overline{BD} \cong \overline{BD}$	Reflexive
$\angle ADB \cong \angle CBD$	AIA
$\triangle ABD \cong \triangle CDB$	ASA
$\overline{AD} \cong \overline{BC}$	CPCTC

Answer:

The completed proof table is as shown above, and the key steps use properties of parallel lines (AIA), reflexive property, ASA congruence, and CPCTC to prove $\overline{AD} \cong \overline{BC}$.