Question

directions: complete each proof using the most appropriate method. (sss or sas)
9 given: \\(\overline{ab} \cong \overline{cd}\\), \\(\overline{ab} \parallel \overline{cd}\\), \\(d\\) is the midpoint of \\(\overline{be}\\)
prove: \\(\triangle abd \cong \triangle cde\\)
(image of triangles abd and cde with b, d, e colinear)
statements | reasons

1. \\(\overline{ab} \cong \overline{cd}\\) | 1. given
2. \\(\overline{ab} \parallel \overline{cd}\\) | 2. given
3. \\(d\\) is the midpoint of \\(\overline{be}\\) | 3. given
4. | 4.
5. | 5.
6. | 6.

Explanation:

Step1: Identify Alternate Interior Angles

Since \( \overline{AB} \parallel \overline{CD} \) and \( \overline{BD} \) and \( \overline{DE} \) are transversals, \( \angle ABD \cong \angle CDE \) (Alternate Interior Angles Theorem).

Step2: Use Midpoint Definition

Because \( D \) is the midpoint of \( \overline{BE} \), \( \overline{BD} \cong \overline{DE} \) (Definition of Midpoint).

Step3: Apply SAS Congruence

We have \( \overline{AB} \cong \overline{CD} \) (Given), \( \angle ABD \cong \angle CDE \) (Step 1), and \( \overline{BD} \cong \overline{DE} \) (Step 2). By the SAS (Side - Angle - Side) Congruence Postulate, \( \triangle ABD \cong \triangle CDE \).

Now, filling in the proof table:

Statements	Reasons
2. \( \overline{AB} \parallel \overline{CD} \)	2. Given
3. \( D \) is the midpoint of \( \overline{BE} \)	3. Given
4. \( \angle ABD \cong \angle CDE \)	4. Alternate Interior Angles Theorem (since \( \overline{AB} \parallel \overline{CD} \))
5. \( \overline{BD} \cong \overline{DE} \)	5. Definition of Midpoint (since \( D \) is the midpoint of \( \overline{BE} \))
6. \( \triangle ABD \cong \triangle CDE \)	6. SAS Congruence Postulate (using statements 1, 4, 5)

Answer:

The completed proof has the following in the table:

1. \( \angle ABD \cong \angle CDE \); 4. Alternate Interior Angles Theorem
2. \( \overline{BD} \cong \overline{DE} \); 5. Definition of Midpoint
3. \( \triangle ABD \cong \triangle CDE \); 6. SAS Congruence Postulate