Question

1. given \\( \overline { b c } \cong \overline { d a } \\) \\( a c \\) bisects \\( \angle b c d \\) prove: \\( \triangle a b c \cong \triangle c d a \\) 1. \\( \overline { b c } \cong \overline { d a } \\) 2. \\( \overline { a c } \\) bisects \\( \angle b c d \\) 3. \\( \angle b c a \cong \angle d c a \\) 4. \\( \overline { a c } \cong \overline { a c } \\) 5. \\( \triangle a b c \cong \triangle c d a \\)

Explanation:

Step1: Given

Given \( \overline{BC}\cong\overline{DA} \)

Step2: Given

Given \( \overline{AC} \) bisects \( \angle BCD \)

Step3: Definition of angle bisector

If a ray bisects an angle, it divides the angle into two congruent angles. So, \( \angle BCA\cong\angle DCA \)

Step4: Reflexive property

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

Step5: SAS (Side - Angle - Side) congruence

In \( \triangle ABC \) and \( \triangle CDA \), we have \( \overline{BC}\cong\overline{DA} \) (side), \( \angle BCA\cong\angle DCA \) (angle), \( \overline{AC}\cong\overline{AC} \) (side). By the SAS (Side - Angle - Side) congruence criterion, \( \triangle ABC\cong\triangle CDA \)

Answer:

1. Given
2. Given
3. Definition of angle bisector
4. Reflexive property of congruence
5. SAS (Side - Angle - Side) congruence criterion