Question

1. given: \\( \overline { a c } \\) bisects \\( \angle b a d \\) \\& \\( \overline { a b } \cong \overline { a d } \\) prove: \\( \triangle a b c \cong \triangle a d c \\) 20)prove that triangles are congruent 21) given: pt. \\( c \\) is the midpoint of \\( \overline { a e } \\). \\( \angle a \cong \angle e \\) prove: \\( \overline { a b } \cong \overline { d e } \\) 22. given: \\( \overline { s r } \cong \overline { u t } ; \overline { s r } \parallel \overline { u t } \\); \\( \angle s \cong \angle u \\) prove: \\( \overline { s t } \parallel \overline { u v } \\) proof: statements reasons 1. \\( \overline { s r } \cong \overline { u t } ; \overline { s r } \parallel \overline { u t } \\); \\( \angle s \cong \angle u \\) 1. 2. \\( \angle 1 \cong \angle 4 \\) 2. 3. \\( \triangle r s t \cong \triangle t u v \\) 3. 4. \\( \angle 3 \cong \angle 2 \\) 4. 5. \\( \overline { s t } \parallel \overline { u v } \\) 5.

Explanation:

Step1: Use the definition of angle bisector

Since \( \overline{AC} \) bisects \( \angle BAD \), we have \( \angle BAC=\angle DAC \).

Step2: Use the Side - Angle - Side (SAS) congruence criterion

We are given \( \overline{AB}\cong\overline{AD} \), \( \angle BAC = \angle DAC \) (from Step 1), and \( \overline{AC}\cong\overline{AC} \) (common side).
By the SAS congruence criterion, in \( \triangle ABC \) and \( \triangle ADC \), if \( AB = AD\), \( \angle BAC=\angle DAC\), and \( AC = AC\), then \( \triangle ABC\cong\triangle ADC \).

Answer:

\( \triangle ABC\cong\triangle ADC \) (by SAS congruence criterion)