Question

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1. given that \\( \overline { a b } \parallel \overline { c d } \\) and \\( \overline { a b } \cong \overline { c d } \\), how can

you show that \\( \angle b \cong \angle d \\)?
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Explanation:

Step1: Connect \(AC\)

Since \(\overline{AB}\parallel\overline{CD}\), \(\angle BAC\cong\angle DCA\) (alternate - interior angles).

Step2: Prove \(\triangle ABC\cong\triangle CDA\)

We know \(\overline{AB}\cong\overline{CD}\), \(\angle BAC\cong\angle DCA\), and \(\overline{AC}\cong\overline{CA}\) (common side). By the SAS (Side - Angle - Side) congruence criterion, \(\triangle ABC\cong\triangle CDA\).

Step3: Conclude \(\angle B\cong\angle D\)

Corresponding parts of congruent triangles are congruent. So, \(\angle B\cong\angle D\) as they are corresponding angles of \(\triangle ABC\) and \(\triangle CDA\).

Answer:

Connect \(AC\). Since \(\overline{AB}\parallel\overline{CD}\), \(\angle BAC\cong\angle DCA\) (alternate - interior angles). Also, \(\overline{AB}\cong\overline{CD}\) and \(\overline{AC}\cong\overline{CA}\) (common side). By SAS, \(\triangle ABC\cong\triangle CDA\). Then, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(\angle B\cong\angle D\).