Question

question
given: \\( \angle b \cong \angle d \\) and \\( \overline{bc} \parallel \overline{ad} \\).
prove: \\( \overline{ab} \cong \overline{cd} \\).

step
statement
reason
1
\\( \angle b \cong \angle d \\)
given
\\( \overline{bc} \parallel \overline{ad} \\)

Explanation:

Step1: Use parallel line property

Since \(\overline{BC}\parallel\overline{AD}\), then \(\angle BCA\cong\angle DAC\) (alternate - interior angles).

Step2: Consider triangle congruence

In \(\triangle ABC\) and \(\triangle CDA\):

• \(\angle B\cong\angle D\) (given)
• \(\angle BCA\cong\angle DAC\) (from Step 1)
• \(AC = CA\) (reflexive property of congruence, \(a=a\))

By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle ABC\cong\triangle CDA\).

Step3: Use congruent triangle property

If \(\triangle ABC\cong\triangle CDA\), then \(\overline{AB}\cong\overline{CD}\) (corresponding parts of congruent triangles are congruent).

Answer:

\(\overline{AB}\cong\overline{CD}\) is proved as above.