Question

1. use a 2 - column or flow chart proof

given: \\( \angle a c b \cong \angle d c b \\) and \\( \overline{a c} \cong \overline{d c} \\)
prove: \\( \overline{c b} \\) bisects \\( \angle a b d \\)

Explanation:

Step1: Identify the common side

In \(\triangle ACB\) and \(\triangle DCB\), \(\overline{CB}\) is common. So, \(\overline{CB}\cong\overline{CB}\) (Reflexive property of congruence).

Step2: Prove the triangles congruent

Given \(\angle ACB\cong\angle DCB\) and \(\overline{AC}\cong\overline{DC}\), by the Side - Angle - Side (SAS) congruence criterion, \(\triangle ACB\cong\triangle DCB\).

Step3: Use congruent triangles to find angle equality

Since \(\triangle ACB\cong\triangle DCB\), then \(\angle ABC\cong\angle DBC\) (Corresponding parts of congruent triangles are congruent).

Step4: Define angle bisector

If \(\angle ABC\cong\angle DBC\), then \(\overline{CB}\) bisects \(\angle ABD\) (Definition of an angle bisector: A ray that divides an angle into two congruent adjacent angles).

Answer:

Since \(\triangle ACB\cong\triangle DCB\) (by SAS) and \(\angle ABC\cong\angle DBC\) (CPCTC), \(\overline{CB}\) bisects \(\angle ABD\) (by the definition of an angle bisector).