Question

given: \\( \overline { b d } \\) bisects \\( \angle a b c \\) and \\( \overline { a b } \cong \overline { b c } \\).
prove: \\( \overline { b d } \\) bisects \\( \angle a d c \\).
note: quadrilateral properties are not permitted in this proof.
step
statement
reason
1
\\( b d \\) bisects \\( \angle a b c \\)
\\( \overline { a b } \cong \overline { b c } \\)
given

Explanation:

Step1: Use the definition of angle bisector

Since \( \overline{BD} \) bisects \( \angle ABC \), we have \( \angle ABD=\angle CBD \).

Step2: Prove \( \triangle ABD\cong\triangle CBD \)

We know \( \overline{AB}\cong\overline{BC} \) (given), \( \angle ABD = \angle CBD \) (from Step 1), and \( \overline{BD}\cong\overline{BD} \) (reflexive property of congruence). By the Side - Angle - Side (SAS) congruence criterion, \( \triangle ABD\cong\triangle CBD \).

Step3: Use the property of congruent triangles

If \( \triangle ABD\cong\triangle CBD \), then \( \angle ADB=\angle CDB \) (corresponding parts of congruent triangles are congruent).

Step4: Use the definition of angle bisector

Since \( \angle ADB=\angle CDB \), by the definition of an angle bisector, \( \overline{BD} \) bisects \( \angle ADC \).

Answer:

\( \overline{BD} \) bisects \( \angle ADC \) is proved.