question\ngiven: \\( \\overline { c a } \\) bisects \\( \\angle b a d \\) and \\( \\overline { a d } \\cong…

question\ngiven: \\( \\overline { c a } \\) bisects \\( \\angle b a d \\) and \\( \\overline { a d } \\cong \\overline { a b } \\).\nprove: \\( \\triangle a b c \\cong \\triangle a d c \\).\n

question\ngiven: \\( \\overline { c a } \\) bisects \\( \\angle b a d \\) and \\( \\overline { a d } \\cong \\overline { a b } \\).\nprove: \\( \\triangle a b c \\cong \\triangle a d c \\).\n

Answer

Explanation:

Step1: Definition of angle bisector

Since (\overline{CA}) bisects (\angle BAD), we have (\angle BAC=\angle DAC) (by the definition of an angle bisector).

Step2: Common side

(\overline{AC}\cong\overline{AC}) (by the reflexive property of congruence, which states that any segment is congruent to itself).

Step3: SAS (Side - Angle - Side) criterion

We are given (\overline{AD}\cong\overline{AB}), we found (\angle BAC = \angle DAC), and (\overline{AC}\cong\overline{AC}). So, by the SAS (Side - Angle - Side) congruence criterion, (\triangle ABC\cong\triangle ADC).

Answer:

(\triangle ABC\cong\triangle ADC) by the SAS (Side - Angle - Side) congruence criterion.