Question

complete the proofs below using the most appropriate method, sss or sas.

1. given: \\( \overline{ab} \cong \overline{ed} \\), \\( \overline{ab} \parallel \overline{de} \\), \\( c \\) is the midpoint of \\( \overline{ae} \\)

prove: \\( \triangle abc \cong \triangle edc \\)
(image of triangle diagram with points a, b, c, d, e)
statements | reasons

1. blank | blank
2. blank | blank
3. blank | blank
4. blank | blank
5. blank | blank
6. blank | blank

(options: given, definition of midpoint, reflexive property, vertical angles, definition of angle bisector, alternate interior angles, alternate exterior angles, corresponding angles, sss, sas, \\( ab \cong ed \\), \\( ac \cong ec \\), \\( ab \parallel de \\), \\( c \\) is the midpoint of \\( ae \\), \\( \angle a \cong \angle e \\), \\( \triangle abc \cong \triangle edc \\))

Explanation:

Step1: Identify Given Info

We start by listing the given information. So, statement 1: $\overline{AB} \cong \overline{ED}$, reason: Given; statement 2: $\overline{AB} \parallel \overline{DE}$, reason: Given; statement 3: $C$ is the midpoint of $\overline{AE}$, reason: Given.

Step2: Use Midpoint Definition

Since $C$ is the midpoint of $\overline{AE}$, by the definition of midpoint, $AC \cong EC$. So statement 4: $AC \cong EC$, reason: Definition of Midpoint.

Step3: Alternate Interior Angles

Because $\overline{AB} \parallel \overline{DE}$ and $\overline{AE}$ is a transversal, the alternate interior angles $\angle A$ and $\angle E$ are congruent. So statement 5: $\angle A \cong \angle E$, reason: Alternate Interior Angles.

Step4: Apply SAS Congruence

Now, we have $\overline{AB} \cong \overline{ED}$ (given), $\angle A \cong \angle E$ (alternate interior angles), and $AC \cong EC$ (midpoint definition). This satisfies the SAS (Side - Angle - Side) congruence criterion. So statement 6: $\triangle ABC \cong \triangle EDC$, reason: SAS.

Answer:

Statements	Reasons
2. $\overline{AB} \parallel \overline{DE}$	Given
3. $C$ is the midpoint of $\overline{AE}$	Given
4. $AC \cong EC$	Definition of Midpoint
5. $\angle A \cong \angle E$	Alternate Interior Angles
6. $\triangle ABC \cong \triangle EDC$	SAS