Question

given bd bisects \\( \angle abc \\); complete the flowchart proof below.
bd bisects \\( \angle abc \\)
\\( \angle abd \cong \angle cbd \\)
\\( \angle a \cong \angle c \\)
\\( \overline { a b } \cong \overline { c b } \\)
\\( \triangle a b d \cong \triangle c b d \\)

Explanation:

Step1: Definition of Angle Bisector

By the definition of an angle bisector, if \(BD\) bisects \(\angle ABC\), then \(\angle ABD\cong\angle CBD\).

Step2: Given Information

From the diagram, we are given that \(AB\cong CB\) (sides of the triangles \(ABD\) and \(CBD\)).

Step3: Angle - Side - Angle (ASA) Criterion

In \(\triangle ABD\) and \(\triangle CBD\):

• \(\angle ABD\cong\angle CBD\) (from the angle - bisector definition)
• \(AB\cong CB\) (given)
• \(BD = BD\) (common side, so \(BD\cong BD\) by the reflexive property of congruence)

By the Angle - Side - Angle (ASA) congruence criterion, \(\triangle ABD\cong\triangle CBD\).

Answer:

For \(\angle ABD\cong\angle CBD\), the reason is "Definition of Angle Bisector". For \(\triangle ABD\cong\triangle CBD\), the reason is "ASA (Angle - Side - Angle) Congruence Criterion".