Question

problem 23
given: \\( \angle b \cong \angle d \\) and
\\( \angle a \cong \angle c \\)
prove: \\( \overline{ab} \cong \overline{dc} \\)

Explanation:

Step1: Identify the triangles

We can consider triangles \( \triangle ABC\) and \( \triangle CDA\).

Step2: Use the Angle - Angle (AA) similarity criterion

Since \( \angle B\cong\angle D\) and \( \angle A\cong\angle C\), and the sum of angles in a triangle is \(180^{\circ}\), we know that the third pair of angles \( \angle BAC\) and \( \angle DCA\) (or other remaining angle pairs depending on triangle naming) are also congruent. But for congruence, if we assume \(AC = AC\) (common side).

Step3: Apply the Angle - Angle - Side (AAS) congruence criterion

In \( \triangle ABC\) and \( \triangle CDA\), \( \angle B\cong\angle D\), \( \angle A\cong\angle C\), and \(AC = AC\) (reflexive property). So, \( \triangle ABC\cong\triangle CDA\) by AAS.

Step4: Conclude the side congruence

Since \( \triangle ABC\cong\triangle CDA\), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) property, \( \overline{AB}\cong\overline{DC}\).

Answer:

1. \( \triangle ABC\) and \( \triangle CDA\) are considered; Reason: For proving congruence.
2. \( \triangle ABC\cong\triangle CDA\) (by AAS, with \(AC = AC\) as the common side); Reason: AAS (Angle - Angle - Side) congruence criterion.

Final conclusion: \( \overline{AB}\cong\overline{DC}\) (by CPCTC).