Question

the proof \\( \triangle abc\cong\triangle dcb \\) that is shown.
what is the missing reason in the proof?

1. \\( \angle a\cong\angle d \\)
2. given
3. \\( cd\parallel ab \\)
4. given
5. \\( cb\cong bc \\)
6. refl. prop.
7. \\( \angle abc\cong\angle dcb \\)
8. alt. int. \\( \angle s \\) are \\( \cong \\)
9. \\( \triangle abc\cong\triangle dcb \\)
10. ?

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

Explanation:

Step1: Recall triangle congruence theorems

We have two angles (\(\angle A\cong\angle D\), \(\angle ABC\cong\angle DCB\)) and one side (\(CB = BC\)) in the two triangles \(\triangle ABC\) and \(\triangle DCB\).

Step2: Identify the congruence theorem

The AAS (Angle - Angle - Side) theorem 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. Here, \(\angle A\cong\angle D\), \(\angle ABC\cong\angle DCB\) and \(CB\cong BC\) (reflexive property) satisfy the AAS condition for \(\triangle ABC\) and \(\triangle DCB\).

Answer:

AAS