Question: Given $\overline{AC} \perp \overline{BD}$, complete the flowchart proof below. Note that the last statement and reason have both been filled in for you.
$B$
$A$
$D$
$C$
The diagram shows triangle $ABC$ with $A$ at the bottom left, $C$ at the bottom right, and $B$ at the top. Segment $BD$ is drawn from $B$ straight down to point $D$ on $AC$, and right-angle/perpendicular relationship is indicated by $\overline{AC} \perp \overline{BD}$. The slanted sides $AB$ and $BC$ are marked congruent.
position_hint=centertop-left: $\overline{AC} \perp \overline{BD}$
Reason:
Given
center: $\overline{AD} \cong \overline{DC}$
Reason:
A midpoint divides a
segment into two
congruent segments
bottom-left: $\triangle ABD$ and
$\triangle CBD$ are right
triangles
Reason:
The triangles each
contain a right angle
center: Type of Statement
Reason:
Given
bottom-right: $\overline{BD} \cong \overline{BD}$
Reason:
Reflexive Property
1. **State the problem:** We are given that $\overline{AC} \perp \overline{BD}$ and need to complete the flowchart proof showing congruence between triangles $\triangle ABD$ and $\triangle CBD$.
2. **Given:** $\overline{AC} \perp \overline{BD}$ means $\angle ADB$ and $\angle CDB$ are right angles.
3. **Midpoint:** $D$ is the midpoint of $\overline{AC}$, so by definition, $\overline{AD} \cong \overline{DC}$.
4. **Right triangles:** Since $\overline{AC} \perp \overline{BD}$, triangles $\triangle ABD$ and $\triangle CBD$ each contain a right angle at $D$.
5. **Reflexive property:** $\overline{BD} \cong \overline{BD}$ because any segment is congruent to itself.
6. **Congruence of triangles:** By the Hypotenuse-Leg (HL) theorem for right triangles, since $\overline{AB} \cong \overline{BC}$ (given), $\overline{BD} \cong \overline{BD}$ (reflexive), and both triangles have a right angle, we conclude $\triangle ABD \cong \triangle CBD$.
7. **Final statement and reason:** $\triangle ABD \cong \triangle CBD$ because of the HL theorem.
This completes the flowchart proof.