Question

given: ( angle acd cong angle bcd ), ( overline{eb} parallel overline{ac} ) and ( overline{ab} perp overline{ce} ).
prove: ( overline{cd} cong overline{de} ).

Explanation:

Step1: Use the property of parallel lines

Since \(EB\parallel AC\), then \(\angle ACD=\angle BED\) (alternate - interior angles).

Step2: Use the given angle equality

Given \(\angle ACD = \angle BCD\), so \(\angle BCD=\angle BED\).

Step3: Use the right - angle property

Because \(AB\perp CE\), \(\angle CDB=\angle EDB = 90^{\circ}\).

Step4: Prove triangles congruent

In \(\triangle CDB\) and \(\triangle EDB\), \(\angle BCD=\angle BED\), \(\angle CDB=\angle EDB\), and \(BD = BD\) (common side). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle CDB\cong\triangle EDB\).

Step5: Conclude the segment equality

Since \(\triangle CDB\cong\triangle EDB\), then \(CD\cong DE\) (corresponding parts of congruent triangles are congruent).

Answer:

\(CD\cong DE\) is proved.