Question

question
given: \\( \overline { c a } \\) bisects \\( \angle b a d \\) and \\( \overline { a d } \cong \overline { a b } \\).
prove: \\( \triangle a b c \cong \triangle a d c \\).

Explanation:

Step1: Use the angle - bisector definition

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

Step2: Consider the common side

\(\overline{AC}\cong\overline{AC}\) (reflexive property of congruence).

Step3: Apply the SAS (Side - Angle - Side) congruence criterion

We are given \(\overline{AD}\cong\overline{AB}\), we have \(\angle BAC = \angle DAC\) (from Step 1), and \(\overline{AC}\cong\overline{AC}\) (Step 2).
By the SAS congruence criterion, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Answer:

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