Question

complete the flowchart proof below.
\\(\overline{ac}\\) bisects \\(\angle bcd\\)
reason:
given
\\(\angle bca \cong \angle dca\\)
reason:
select reason
\\(\angle b \cong \angle d\\)
reason:
select reason
\\(\overline{ac} \cong \overline{ac}\\)
reason:
select reason
\\(\triangle abc \cong \triangle adc\\)
reason:
select reason

Explanation:

Step1: Definition of angle bisector

If a ray bisects an angle, it divides the angle into two congruent angles. Since \( \overline{AC} \) bisects \( \angle BCD \), then \( \angle BCA\cong\angle DCA \).

Step2: Given information

From the diagram, we can see that \( \angle B\cong\angle D \) (given in the problem - likely from the figure's markings).

Step3: Reflexive property of congruence

For any segment \( \overline{AC} \), \( \overline{AC}\cong\overline{AC} \) (a segment is congruent to itself).

Step4: AAS (Angle - Angle - Side) congruence criterion

In \( \triangle ABC \) and \( \triangle ADC \), we have \( \angle BCA\cong\angle DCA \), \( \angle B\cong\angle D \), and \( \overline{AC}\cong\overline{AC} \). By the AAS (Angle - Angle - Side) congruence rule, \( \triangle ABC\cong\triangle ADC \).

Answer:

• For \( \angle BCA\cong\angle DCA \), the reason is "Definition of angle bisector".
• For \( \angle B\cong\angle D \), the reason is "Given".
• For \( \overline{AC}\cong\overline{AC} \), the reason is "Reflexive property of congruence".
• For \( \triangle ABC\cong\triangle ADC \), the reason is "AAS (Angle - Angle - Side)".