Question

the proof \\( \triangle a b c \cong \triangle d c b \\) that is shown.
given: \\( \angle a \cong \angle d ; \overline{c d} \\| \overline{a b} \\)
prove: \\( \triangle a b c \cong \triangle d c b \\)
what is the missing reason in the proof?
\\( \begin{array} { c | c } { \text { statement } } & { \text { reason } } \\\\ hline 1. & { \text { given } } \\\\ 2. & { \text { given } } \\\\ 3. & { \text { refl. prop. } } \\\\ 4. & { \text { alt. int. } angle s \text { are } cong } \\\\ 5. & {? } end{array} \\)
\\( \bigcirc \\) alt. ext. \\( \angle s \\) are \\( \cong \\)
\\( \bigcirc \\) asa
\\( \bigcirc \\) aas
\\( \bigcirc \\) corr. int. \\( \angle s \\) are \\( \cong \\)

Explanation:

• In the proof, we have two angles (\(\angle A\cong\angle D\) and \(\angle ABC\cong\angle DCB\)) and a non - included side (\(CB\cong BC\)).
• The AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.
• Let's check the other options:
• "alt. ext. \(\angle s\) are \(\cong\)" is about alternate exterior angles (not relevant to triangle congruence proof here).
• ASA (Angle - Side - Angle) requires the side to be included between the two angles. Here \(CB\) is not included between \(\angle A\) and \(\angle ABC\) (or \(\angle D\) and \(\angle DCB\)).
• "corr. int. \(\angle s\) are \(\cong\)" is about corresponding interior angles (not relevant to triangle congruence proof here).

Answer:

AAS