Question

the proof \\(\delta abc \cong \delta dcb\\) that is shown.

given: \\(\angle a \cong \angle d\\); \\(\overline{cd} \parallel \overline{ab}\\)
prove: \\(\delta abc \cong \delta dcb\\)

what is the missing reason in the proof?

\

$$\begin{tabular}{l|l} statement & reason \\\\ \\hline 1. \\(\\angle a \\cong \\angle d\\) & 1. given \\\\ 2. \\(\\overline{cd} \\parallel \\overline{ab}\\) & 2. given \\\\ 3. \\(\\overline{cb} \\cong \\overline{bc}\\) & 3. refl. prop. \\\\ 4. \\(\\angle abc \\cong \\angle dcb\\) & 4. alt. int. \\(\\angle\\)s are \\(\\cong\\) \\\\ 5. \\(\\delta abc \\cong \\delta dcb\\) & 5. ? \\end{tabular}$$

alt. ext. \\(\angle\\)s are \\(\cong\\)
asa
aas
corr. int. \\(\angle\\)s are \\(\cong\\)

Explanation:

Identify the established congruent parts

Using the Triangle Congruence Proofs knowledge point

• Statement 1: \(\angle A \cong \angle D\) (Angle)
• Statement 3: \(\overline{CB} \cong \overline{BC}\) (Side)
• Statement 4: \(\angle ABC \cong \angle DCB\) (Angle)

Determine the congruence theorem

Using the AAS Congruence Theorem knowledge point

• The congruent parts are two angles and a non-included side: \(\angle A\), \(\angle ABC\), and side \(\overline{BC}\) for \(\triangle ABC\); and \(\angle D\), \(\angle DCB\), and side \(\overline{CB}\) for \(\triangle DCB\).
• Therefore, the triangles are congruent by the Angle-Angle-Side (AAS) congruence theorem.

Answer:

• alt. ext. \(\angle\)s are \(\cong\)
• ASA
• AAS (Correct answer)
• corr. int. \(\angle\)s are \(\cong\)