Question

which piece of additional information can be used to prove \\( \triangle cea - \triangle cdb \\)?
\\( \angle bdc \\) and \\( \angle aed \\) are right angles
\\( \overline { ae } \cong \overline { ed } \\)
\\( \triangle bdc \\) is a right triangle
\\( \angle dbc \cong \angle dcb \\)

Explanation:

Step1: Recall similarity criteria

For two triangles to be similar, we can use AA (Angle - Angle) similarity criterion.

Step2: Analyze each option

• If $\angle BDC$ and $\angle AED$ are right angles, then $\angle AEC=\angle BDC = 90^{\circ}$. Also, $\angle C$ is common to both $\triangle CEA$ and $\triangle CDB$. So, by AA similarity ($\angle C=\angle C$ and $\angle AEC=\angle BDC$), $\triangle CEA\sim\triangle CDB$.
• If $\overline{AE}\cong\overline{ED}$, this gives information about side - length equality in $\triangle AED$ but no information about angles or ratios of sides for similarity of $\triangle CEA$ and $\triangle CDB$.
• If $\triangle BDC$ is a right triangle, we don't know which angle is the right angle. If we don't know that it has an angle equal to an angle in $\triangle CEA$, we can't prove similarity.
• If $\angle DBC\cong\angle DCB$, this gives information about $\triangle BDC$ being isosceles ($BD = CD$) but no information about angles related to $\triangle CEA$ for similarity.

Answer:

$\angle BDC$ and $\angle AED$ are right angles