Question

given ( d ) is the midpoint of ( overline{ac} ) and ( overline{ac} perp overline{bd} ), complete the flowchart proof below.

( overline{ac} perp overline{bd} )
reason: given

( angle adb ) and ( angle cdb ) are right angles
reason: perpendicular lines form right angles

( d ) is the midpoint of ( overline{ac} )
reason: given

( angle adb cong angle cdb )
reason:

( overline{ad} cong overline{cd} )
reason:

( overline{bd} cong overline{bd} )
reason:

( \triangle abd cong \triangle cbd )
reason:

Explanation:

Step1: Determine the reason for $\angle ADB\cong\angle CDB$

All right angles are congruent. Since $\angle ADB$ and $\angle CDB$ are right angles (from $AC\perp BD$), the reason is "All right angles are congruent".

Step2: Determine the reason for $\overline{AD}\cong\overline{CD}$

By the definition of a midpoint, if $D$ is the midpoint of $\overline{AC}$, then $AD = CD$, so $\overline{AD}\cong\overline{CD}$ (Reason: "Definition of midpoint").

Step3: Determine the reason for $\overline{BD}\cong\overline{BD}$

A segment is congruent to itself. This is the reflexive property of congruence. So the reason is "Reflexive property of congruence".

Step4: Determine the reason for $\triangle ABD\cong\triangle CBD$

We have $\angle ADB\cong\angle CDB$ (right angles), $\overline{AD}\cong\overline{CD}$ (from mid - point), and $\overline{BD}\cong\overline{BD}$ (reflexive property). By the Side - Angle - Side (SAS) congruence criterion, $\triangle ABD\cong\triangle CBD$.

Answer:

• For $\angle ADB\cong\angle CDB$: All right angles are congruent.
• For $\overline{AD}\cong\overline{CD}$: Definition of midpoint.
• For $\overline{BD}\cong\overline{BD}$: Reflexive property of congruence.
• For $\triangle ABD\cong\triangle CBD$: SAS (Side - Angle - Side) congruence criterion.