Question

select reason
reason:
$overline{ad}congoverline{cd}$
reason: given
$overline{bd}$ bisects $overline{ac}$
select reason
reason:
$angle acongangle c$
select reason
reason:
$overline{ab}congoverline{cb}$
$\triangle abdcong\triangle cbd$

Explanation:

Step1: Determine the reason for \( \overline{AD}\cong\overline{CD} \)

Since \( BD \) bisects \( AC \), by the definition of a bisector, it divides \( AC \) into two equal parts. So, \( AD = CD \), which means \( \overline{AD}\cong\overline{CD} \) (Definition of Segment Bisector).

Step2: Determine the reason for \( \angle A\cong\angle C \)

This is not directly given from the " \( BD \) bisects \( AC \)" statement. There is no information in the provided "given" (only \( BD \) bisects \( AC \)) to support this congruence of angles. If we assume it's part of a triangle - related property (but no triangle information is given in a way to justify this from the "given" here). However, if we consider the final goal of proving \( \triangle ABD\cong\triangle CBD \), and if we assume some missing context (like maybe it's an isosceles - related figure in a more complete problem setup), but from the given " \( BD \) bisects \( AC \)" alone, there is no valid geometric reason. But if we assume it's a typo and it should be related to the bisector (but no). If we consider the answer - seeking for a common geometric proof (maybe in a SAS - like proof for the triangles), but with the given, we can't. But if we assume it's a mis - written and should be related to the bisector's property on angles (but no). So, if we consider the most probable in a basic proof (assuming some standard textbook problem where with \( BD \) bisecting \( AC \) (so \( AD = CD \), \( AB = CB \) (from bisector property in some figures, say if \( \triangle ABC \) is isosceles with \( BD \) as median and altitude and angle - bisector, but with the given \( BD \) bisects \( AC \), if \( \triangle ABC \) is isosceles \( AB = CB \)) \( \angle A\cong\angle C \) (Base Angles Theorem: In an isosceles triangle, the angles opposite the equal sides are equal. If \( AB = CB \), then \( \angle A\cong\angle C \)).

Step3: Determine the reason for \( \overline{AB}\cong\overline{CB} \)

If \( BD \) bisects \( AC \) ( \( AD = CD \)) and we assume (from the goal of proving \( \triangle ABD\cong\triangle CBD \)) and using a congruence postulate (say SSS or SAS). If we consider the Base Angles Theorem converse (if \( \angle A\cong\angle C \), then \( AB = CB \)) or if \( BD \) is also an altitude and angle - bisector (by the Isosceles Triangle Theorem: If a line bisects the base of an isosceles triangle, then it is also the altitude and angle - bisector. But with only \( BD \) bisects \( AC \) given, if we assume the triangle is isosceles (a common textbook problem setup), then \( AB = CB \) (Definition of Isosceles Triangle: A triangle with two equal sides).

Step4: Determine the reason for \( \triangle ABD\cong\triangle CBD \)

We have \( \overline{AD}\cong\overline{CD} \), \( \overline{AB}\cong\overline{CB} \), and \( \overline{BD}\cong\overline{BD} \) (Reflexive Property: A segment is congruent to itself). So, by SSS (Side - Side - Side) Congruence Postulate: If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. So \( \triangle ABD\cong\triangle CBD\) (SSS).

Answer:

• For \( \overline{AD}\cong\overline{CD} \): Definition of Segment Bisector.
• For \( \angle A\cong\angle C \): Base Angles Theorem (assuming \( AB = CB \) from triangle being isosceles).
• For \( \overline{AB}\cong\overline{CB} \): Definition of Isosceles Triangle (assuming \( \triangle ABC \) is isosceles).
• For \( \triangle ABD\cong\triangle CBD \): SSS (Side - Side - Side) Congruence Postulate.