Question

given: \\(\overline{ac}\\) bisects \\(\angle dab\\) and \\(\angle dcb\\).
prove: \\(\triangle abc \cong \triangle adc\\)

1. \\(\overline{ac}\\) bisects \\(\angle dab\\) and \\(\angle dcb\\) .
2. \\(\angle dab \cong \angle bac\\)
3. \\(\angle dca \cong \angle bca\\)
4. \\(\overline{ac} \cong \overline{ac}\\)
5. \\(\triangle abc \cong \triangle adc\\)

\\(\square\\) a) sss
\\(\square\\) b) asa
\\(\square\\) c) sas
\\(\square\\) d) given
\\(\square\\) e) definition of angle bisector
\\(\square\\) f) definition of congruent segments
\\(\square\\) g) reflexive property of congruence

Explanation:

Step1: Analyze Step 2

To determine the reason for \( \angle DAB \cong \angle BAC \), we recall the definition of an angle bisector. An angle bisector divides an angle into two congruent angles. Since \( \overline{AC} \) bisects \( \angle DAB \), it splits \( \angle DAB \) into \( \angle DAC \) (wait, correction: the step 2 should be \( \angle DAC \cong \angle BAC \) probably a typo, but following the logic, the reason is the definition of angle bisector. So for step 2, the reason is E) Definition of angle bisector.

Step2: Analyze Step 3

Similarly, \( \overline{AC} \) bisects \( \angle DCB \), so it splits \( \angle DCB \) into \( \angle DCA \) and \( \angle BCA \), making them congruent. The reason for \( \angle DCA \cong \angle BCA \) is also E) Definition of angle bisector.

Step3: Analyze Step 4

The statement \( \overline{AC} \cong \overline{AC} \) is true because a segment is congruent to itself, which is the Reflexive Property of Congruence. So the reason for step 4 is G) Reflexive Property of Congruence.

Step4: Analyze Step 5

To prove \( \triangle ABC \cong \triangle ADC \), we have two angles and the included side. \( \angle DAC \cong \angle BAC \) (step 2), \( \overline{AC} \cong \overline{AC} \) (step 4), and \( \angle DCA \cong \angle BCA \) (step 3). This is the ASA (Angle - Side - Angle) congruence criterion, where two angles and the included side of one triangle are congruent to two angles and the included side of another triangle. So the reason for step 5 is B) ASA.

Answer:

For step 2: E. Definition of angle bisector
For step 3: E. Definition of angle bisector
For step 4: G. Reflexive Property of Congruence
For step 5: B. ASA

(Assuming we are matching each step to its reason. If the question was about the congruence postulate for step 5, the answer is B) ASA. If it's about the reasons for each step, the above are the matches.)