Question

complete the following proof.
given:

• c is the midpoint of \\(\overline{ae}\\)
• c is the midpoint of \\(\overline{bd}\\)
• \\(\overline{ab} \cong \overline{ed}\\)

prove: \\(\triangle abc \cong \triangle edc\\)
image of a diagram with points a, c, e on a horizontal line, and points b, c, d with triangles abc and edc, each with a marked congruent side
it is given that c is the midpoint of \\(\overline{ae}\\), c is the midpoint of \\(\overline{bd}\\), and \\(\overline{ab} \cong \overline{ed}\\).
since c is the midpoint of \\(\overline{ae}\\), \\(\underline{\quad\quad\quad\quad}\\) by the definition of \\(\underline{\quad\quad\quad\quad}\\).
since c is the midpoint of \\(\overline{bd}\\), \\(\underline{\quad\quad\quad\quad}\\) by the definition of \\(\underline{\quad\quad\quad\quad}\\).
by all of these congruencies, \\(\triangle abc \cong \triangle edc\\) by the \\(\underline{\quad\quad\quad\quad}\\) triangle congruence theorem.

Explanation:

Step1: Analyze midpoint of \( \overline{AE} \)

A midpoint divides a segment into two congruent parts. So if \( C \) is the midpoint of \( \overline{AE} \), then \( \overline{AC} \cong \overline{EC} \) by the definition of midpoint.

Step2: Analyze midpoint of \( \overline{BD} \)

Similarly, since \( C \) is the midpoint of \( \overline{BD} \), \( \overline{BC} \cong \overline{DC} \) by the definition of midpoint.

Step3: Identify congruence theorem

We know \( \overline{AB} \cong \overline{ED} \) (given), \( \overline{AC} \cong \overline{EC} \) (from midpoint of \( AE \)), and \( \overline{BC} \cong \overline{DC} \) (from midpoint of \( BD \)). So by the Side - Side - Side (SSS) Triangle Congruence Theorem, \( \triangle ABC \cong \triangle EDC \).

Answer:

1. For the first blank (midpoint of \( \overline{AE} \)): \( \overline{AC} \cong \overline{EC} \); midpoint.
2. For the second blank (midpoint of \( \overline{BD} \)): \( \overline{BC} \cong \overline{DC} \); midpoint.
3. For the last blank (congruence theorem): SSS.