Question

which piece of additional information can be used to prove △cea ~ △cdb?
○ ∠bdc and ∠aed are right angles
○ ( overline{ae} cong overline{ed} )
○ △bdc is a right triangle
○ ( angle dbc cong angle dcb ) (with diagram showing points a, e, d, c, b where e-d-c are colinear, a-e is vertical, b-d is vertical, and a-b-c are colinear)

Explanation:

To prove \(\triangle CEA \sim \triangle CDB\), we can use the AA (Angle - Angle) similarity criterion. We already know that \(\angle C\) is common to both triangles. If we can show that another pair of angles is equal (preferably right angles as they are easy to identify), we can prove similarity.

• Option 1: If \(\angle BDC\) and \(\angle AED\) are right angles, then \(\angle BDC=\angle AEC = 90^{\circ}\) (since \(\angle AED\) and \(\angle AEC\) are supplementary? No, actually \(\angle AED\) and \(\angle AEC\) are the same line, so \(\angle AEC = 90^{\circ}\) if \(\angle AED=90^{\circ}\) and \(\angle BDC = 90^{\circ}\). Then we have \(\angle C=\angle C\) (common angle) and \(\angle AEC=\angle BDC = 90^{\circ}\), so by AA similarity, \(\triangle CEA\sim\triangle CDB\).
• Option 2: \(\overline{AE}\cong\overline{ED}\) does not give us any information related to the angles or sides that would help in proving the similarity of \(\triangle CEA\) and \(\triangle CDB\).
• Option 3: Just stating that \(\triangle BDC\) is a right triangle does not tell us if \(\angle AEC\) is also a right triangle. We need to know the relationship between the angles of the two triangles.
• Option 4: \(\angle DBC\cong\angle DCB\) tells us about the angles within \(\triangle BDC\) but does not give us a relationship with the angles of \(\triangle CEA\).

Answer:

\(\boldsymbol{\angle BDC}\) and \(\boldsymbol{\angle AED}\) are right angles