8. given: c is midpoint of $ \overline{bd} $ $ \overline{ab} \perp \overline{bd} $ $ \overline{bd} \perp \overline{de} $ prove: $ \triangle abc \cong \triangle edc $ 1. c is midpoint of $ \overline{bd} $ 2. $ \overline{ab} \perp \overline{bd} $ and $ \overline{bd} \perp \overline{de} $ 3. $ \overline{bc} \cong \overline{cd} $ 4. $ \angle bca \cong \angle ecd $ 5. $ \angle abc $ and $ \angle edc $ are right angles 6. $ \angle abc \cong \angle edc $ 7. $ \triangle abc \cong \triangle edc $

Question

Accepted Answer

# Explanation:
## Step1: Given
Given in the problem statement.
## Step2: Given
Given in the problem statement.
## Step3: Definition of midpoint
Since $C$ is the mid - point of $\overline{BD}$, by the definition of a mid - point ($BC = CD$), so $\overline{BC}\cong\overline{CD}$.
## Step4: Vertical angles
$\angle BCA$ and $\angle ECD$ are vertical angles. By the vertical angles theorem, $\angle BCA\cong\angle ECD$.
## Step5: Definition of perpendicular
Since $\overline{AB}\perp\overline{BD}$ and $\overline{BD}\perp\overline{DE}$, by the definition of perpendicular lines (the angle between them is $90^{\circ}$), $\angle ABC$ and $\angle EDC$ are right angles.
## Step6: All right angles are congruent
Since $\angle ABC$ and $\angle EDC$ are right angles, and all right angles are congruent, $\angle ABC\cong\angle EDC$.
## Step7: ASA (Angle - Side - Angle) congruence criterion
In $\triangle ABC$ and $\triangle EDC$, we have $\angle ABC\cong\angle EDC$, $\overline{BC}\cong\overline{CD}$, $\angle BCA\cong\angle ECD$. By the ASA (Angle - Side - Angle) congruence criterion for triangles, $\triangle ABC\cong\triangle EDC$.

# Answer:
| Statement | Reason |
|--|--|
| 1. $C$ is midpoint of $\overline{BD}$ | Given |
| 2. $\overline{AB}\perp\overline{BD}$ and $\overline{BD}\perp\overline{DE}$ | Given |
| 3. $\overline{BC}\cong\overline{CD}$ | Definition of mid - point |
| 4. $\angle BCA\cong\angle ECD$ | Vertical angles theorem |
| 5. $\angle ABC$ and $\angle EDC$ are right angles | Definition of perpendicular |
| 6. $\angle ABC\cong\angle EDC$ | All right angles are congruent |
| 7. $\triangle ABC\cong\triangle EDC$ | ASA (Angle - Side - Angle) congruence criterion |

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QUESTION IMAGE

Question

Explanation:

Step1: Given

Step2: Given

Step3: Definition of midpoint

Step4: Vertical angles

Step5: Definition of perpendicular

Step6: All right angles are congruent

Step7: ASA (Angle - Side - Angle) congruence criterion

Answer:

Statement	Reason
2. \(\overline{AB}\perp\overline{BD}\) and \(\overline{BD}\perp\overline{DE}\)	Given
3. \(\overline{BC}\cong\overline{CD}\)	Definition of mid - point
4. \(\angle BCA\cong\angle ECD\)	Vertical angles theorem
5. \(\angle ABC\) and \(\angle EDC\) are right angles	Definition of perpendicular
6. \(\angle ABC\cong\angle EDC\)	All right angles are congruent
7. \(\triangle ABC\cong\triangle EDC\)	ASA (Angle - Side - Angle) congruence criterion