Question

question
given \\( \overline { b d } \\) is the segment bisector of \\( \overline { a c } \\), complete the flowchart proof below.
\\( b d \\) bisects \\( a c \\)
reason:
given
\\( \overline { a d } \cong \overline { c d } \\)
\\( \angle a d b \cong \angle c d b \\)
\\( \angle a \cong \angle c \\)
reason:
reason:
reason:
\\( \triangle a b d \cong \triangle c b d \\)
reason:

Explanation:

Step1: Reason for \( \overline{AD}\cong\overline{CD} \)

By the definition of a segment bisector, if a segment \( BD \) bisects another segment \( AC \), then it divides \( AC \) into two congruent parts. So the reason is "Definition of segment bisector".

Step2: Reason for \( \angle ADB\cong\angle CDB \)

Since \( \angle ADB \) and \( \angle CDB \) are linear - pair adjacent angles and their sum is \( 180^{\circ}\), and they are equal in measure (as they are vertical angles formed by the intersection of two lines \( AC \) and \( BD \)), the reason is "Vertical angles are congruent".

Step3: Reason for \( \angle A\cong\angle C \)

These are given in the problem (from the marking on the triangle in the diagram, the angles at \( A \) and \( C \) are marked as equal). So the reason is "Given".

Step4: Reason for \( \triangle ABD\cong\triangle CBD \)

We have \( \overline{AD}\cong\overline{CD}\) (from step 1), \( \angle ADB\cong\angle CDB\) (from step 2), and \( \angle A\cong\angle C\) (from step 3). By the Angle - Side - Angle (ASA) congruence criterion, \( \triangle ABD\cong\triangle CBD\). So the reason is "ASA (Angle - Side - Angle)".

Answer:

• For \( \overline{AD}\cong\overline{CD} \): Definition of segment bisector
• For \( \angle ADB\cong\angle CDB \): Vertical angles are congruent
• For \( \angle A\cong\angle C \): Given
• For \( \triangle ABD\cong\triangle CBD \): ASA (Angle - Side - Angle)