Question

given: in △abc shown, \\(\overline{ba} \cong \overline{bc}\\).
prove: \\(\angle a \cong \angle c\\)

statement	reason
2)	2)
3) \\(\overline{ad} \cong \overline{cd}\\)	3) definition of midpoint
4)	4)
5)	5)
6) \\(\angle a \cong \angle c\\)	6)

complete statements and reasons in the steps:

1. statement? 2. reason?
2. statement? reason?
3. statement? 5. reason?

and
what is the right reason for step 6?

Explanation:

Step1: Add midpoint D to $\overline{AC}$

Let $D$ be the midpoint of $\overline{AC}$, so we draw $\overline{BD}$.

Step2: State $\overline{BD} \cong \overline{BD}$

$\overline{BD} \cong \overline{BD}$

Step3: Reason for Step2: Reflexive property

Reflexive Property of Congruence

Step4: State $\triangle ABD \cong \triangle CBD$

$\triangle ABD \cong \triangle CBD$

Step5: Reason for Step4: SSS congruence

SSS (Side-Side-Side) Congruence Postulate

Step6: Reason for Step6: CPCTC

Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Answer:

Statement	Reason
2) $\overline{BD} \cong \overline{BD}$ (where $D$ is midpoint of $\overline{AC}$)	2) Reflexive Property of Congruence
3) $\overline{AD} \cong \overline{CD}$	3) Definition of midpoint
4) $\triangle ABD \cong \triangle CBD$	4) SSS Congruence Postulate
5) (No additional statement needed; leads directly to step 6)	5) (N/A, supports congruence for step 6)
6) $\angle A \cong \angle C$	6) Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Individual requested steps:

1. Statement: $\boldsymbol{\overline{BD} \cong \overline{BD}}$ (with $D$ as midpoint of $\overline{AC}$); Reason: Reflexive Property of Congruence
2. Statement: $\boldsymbol{\triangle ABD \cong \triangle CBD}$; Reason: SSS Congruence Postulate
3. Statement: (No new statement; this step confirms the triangle congruence leads to angle congruence); Reason: Supports CPCTC for step 6

Reason for step 6: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)