Question

write a paragraph proof.

1. given: \\( \overline { b d } \\) bisects \\( \angle a b c \\) and \\( \angle a d c \\),

\\( \overline { a b } \cong \overline { c b } \\), \\( \overline { a b } \cong \overline { a d } \\), \\( \overline { c b } \cong \overline { d c } \\)
prove: \\( \triangle a b d \cong \triangle c b d \\)

Explanation:

Step1: Use the angle - bisector property

Since \(BD\) bisects \(\angle ABC\) and \(\angle ADC\), we have \(\angle ABD=\angle CBD\) and \(\angle ADB = \angle CDB\).

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

We are given that \(\overline{AB}\cong\overline{CB}\) and \(\overline{BD}\) is common to both \(\triangle ABD\) and \(\triangle CBD\).
In \(\triangle ABD\) and \(\triangle CBD\):

• \(\overline{AB}\cong\overline{CB}\) (given)
• \(\angle ABD=\angle CBD\) (because \(BD\) bisects \(\angle ABC\))
• \(\overline{BD}\cong\overline{BD}\) (common side)

By the SAS congruence criterion, \(\triangle ABD\cong\triangle CBD\)

Answer:

Since \(BD\) bisects \(\angle ABC\) and \(\angle ADC\), \(\angle ABD = \angle CBD\) and \(\angle ADB=\angle CDB\). Also, given \(\overline{AB}\cong\overline{CB}\) and \(\overline{BD}\) is common to both triangles. By the Side - Angle - Side (SAS) congruence criterion (two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle), \(\triangle ABD\cong\triangle CBD\)