Question

the proof below may or may not be correct. if the proof is incorrect, determine the first step number that is not justified and the reason it is not justified. given: \\( \overline { b d } \\) and \\( \overline { a c } \\) bisect each other. prove: \\( \overline { a b } \parallel \overline { c d } \\).

Explanation:

Step1: Check each step

• Step 1: Given, correct.
• Step 2: A segment bisector divides a segment into two congruent segments. Since \(BD\) and \(AC\) bisect each other, \(\overline{AE}\cong\overline{EC}\), correct.
• Step 3: A segment bisector divides a segment into two congruent segments. Since \(BD\) and \(AC\) bisect each other, \(\overline{DE}\cong\overline{EB}\), correct.
• Step 4: Vertical angles are congruent. \(\angle AEB\) and \(\angle CED\) are vertical angles, so \(\angle AEB\cong\angle CED\), correct.
• Step 5: For \(\triangle ABE\) and \(\triangle CDE\), we have \(\overline{AE}\cong\overline{EC}\), \(\angle AEB\cong\angle CED\), \(\overline{DE}\cong\overline{EB}\). By SAS (Side - Angle - Side) congruence criterion, \(\triangle ABE\cong\triangle CDE\), correct.
• Step 6: Corresponding Parts of Congruent Triangles are Congruent (CPCTC). So \(\angle A\cong\angle C\), correct.
• Step 7: If two lines cut by a transversal form congruent alternate interior angles, then the two lines are parallel. Here, \(\overline{AB}\) and \(\overline{CD}\) are cut by transversal \(AC\), and \(\angle A\) and \(\angle C\) are alternate interior angles. Since \(\angle A\cong\angle C\), \(\overline{AB}\parallel\overline{CD}\), correct.

Answer:

The proof is correct.