Question

given $overline{ac}perpoverline{bd}$, complete the flowchart proof below.
$overline{ac}perpoverline{bd}$
reason: given
$angle adb$ and $angle cdb$ are right angles
reason: perpendicular lines form right angles
$angle adbcongangle cdb$
reason:
$overline{ad}congoverline{cd}$
reason:
$angle acongangle c$
reason:
$\triangle abdcong\triangle cbd$
reason:

Explanation:

Step1: Define congruent right - angles

All right angles are congruent. So, if $\angle ADB$ and $\angle CDB$ are right angles, then $\angle ADB\cong\angle CDB$.

Step2: State reason for $AD\cong CD$

There is no information given in the problem - statement about why $AD\cong CD$. But if we assume it is given as an additional condition.

Step3: State reason for $\angle A\cong\angle C$

In $\triangle ABD$ and $\triangle CBD$, since $\angle ADB\cong\angle CDB$, $AD\cong CD$ and $BD = BD$ (common side), by the Side - Angle - Side (SAS) congruence criterion for right - triangles (HL can also be considered if $BD$ is the hypotenuse and $AD, CD$ are legs), $\triangle ABD\cong\triangle CBD$. And corresponding parts of congruent triangles are congruent, so $\angle A\cong\angle C$.

Step4: Prove $\triangle ABD\cong\triangle CBD$

We have $\angle ADB\cong\angle CDB$ (right - angles are congruent), $AD\cong CD$ (assumed given), and $BD$ is common to both triangles. By the Side - Angle - Side (SAS) congruence postulate, $\triangle ABD\cong\triangle CBD$.

Answer:

• $\angle ADB\cong\angle CDB$ Reason: All right angles are congruent.
• $AD\cong CD$ Reason: Given (assumed as no other reason is provided in the problem).
• $\angle A\cong\angle C$ Reason: Corresponding parts of congruent triangles are congruent.
• $\triangle ABD\cong\triangle CBD$ Reason: Side - Angle - Side (SAS) congruence postulate.