Question

directions - complete the proof below (answers may be used more than once):
given: c is the midpoint of \\( \overline { a e } \\) \\( \overline { b c } \cong \overline { d c } \\)
prove: \\( \angle a \cong \angle e \\)
\\( c \\) is the midpoint of \\( \overline { a e } \\) given
\\( \overline { a c } \cong \overline { e c } \\) given
\\( \angle b c a \cong \angle d c e \\) def. of vertical angles
\\( \overline { b c } \cong \overline { d c } \\) given
\\( \triangle a b c \cong \triangle e d c \\) sas

Explanation:

Step1: Midpoint Definition

Since \(C\) is the midpoint of \(\overline{AE}\), by the definition of a midpoint, \(AC = EC\), so \(\overline{AC}\cong\overline{EC}\).

Step2: Vertical Angles

\(\angle BCA\) and \(\angle DCE\) are vertical angles. By the definition of vertical angles, \(\angle BCA\cong\angle DCE\).

Step3: Given Information

We are given that \(\overline{BC}\cong\overline{DC}\).

Step4: SAS Congruence

In \(\triangle ABC\) and \(\triangle EDC\), we have \(\overline{AC}\cong\overline{EC}\), \(\angle BCA\cong\angle DCE\), and \(\overline{BC}\cong\overline{DC}\). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle ABC\cong\triangle EDC\).

Step5: Corresponding Parts of Congruent Triangles

Since \(\triangle ABC\cong\triangle EDC\), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, \(\angle A\cong\angle E\).

Answer:

The proof is completed as shown above to establish that \(\angle A\cong\angle E\).