4.08: swbat prove two triangles are congruent using perpendicular lines
directions: draw and annotate all diagrams and show all work. a quality proof table consisting of statements
and reasons must be created in order to receive full credit.
**#1.) complete the following proof by filling in the blanks. you will be graded on your statements and reasons
and the quality of your annotations.
given: $\overline{de} \perp \overline{ac}$ and $e$ is the midpoint of $\overline{ac}$
prove: $\delta aed \cong \delta ced$
(image of a quadrilateral $abcd$ with diagonals intersecting at $e$)
| statements | reasons |
| --- | --- |
| 1) | 1) |
| 2) $\overline{ae} \cong \overline{ce}$ | 2) |
| 3) | 3) |
| 4) | 4) |
| 5) | 5) |
| 6) $\delta aed \cong \delta ced$ | 6) |

Question

Accepted Answer

# Explanation:
## Step1: Identify Given Information
We know $ \overline{DE} \perp \overline{AC} $ and $ E $ is the midpoint of $ \overline{AC} $. So for statement 1, we state the given: $ \overline{DE} \perp \overline{AC} $ and $ E $ is the midpoint of $ \overline{AC} $. Reason: Given.

## Step2: Midpoint Definition
Since $ E $ is the midpoint of $ \overline{AC} $, by the definition of a midpoint, $ \overline{AE} \cong \overline{CE} $. So reason 2: Definition of Midpoint.

## Step3: Perpendicular Angles
From $ \overline{DE} \perp \overline{AC} $, we know $ \angle AED $ and $ \angle CED $ are right angles. So statement 3: $ \angle AED \cong \angle CED $ (all right angles are congruent). Reason 3: Definition of Perpendicular Lines (they form right angles, and right angles are congruent).

## Step4: Common Side
$ \overline{DE} $ is common to both $ \triangle AED $ and $ \triangle CED $, so statement 4: $ \overline{DE} \cong \overline{DE} $. Reason 4: Reflexive Property of Congruence.

## Step5: SAS Congruence
Now we have $ \overline{AE} \cong \overline{CE} $, $ \angle AED \cong \angle CED $, and $ \overline{DE} \cong \overline{DE} $. So by the Side - Angle - Side (SAS) Congruence Postulate, $ \triangle AED \cong \triangle CED $. Reason 6: SAS Congruence Postulate.

Filling in the table:

| Statements | Reasons |
| --- | --- |
| 1) $ \overline{DE} \perp \overline{AC} $ and $ E $ is the midpoint of $ \overline{AC} $ | 1) Given |
| 2) $ \overline{AE} \cong \overline{CE} $ | 2) Definition of Midpoint |
| 3) $ \angle AED \cong \angle CED $ | 3) Definition of Perpendicular Lines (Right angles are congruent) |
| 4) $ \overline{DE} \cong \overline{DE} $ | 4) Reflexive Property of Congruence |
| 5) (Not needed as we can go to congruence) | 5) (N/A) |
| 6) $ \triangle AED \cong \triangle CED $ | 6) SAS Congruence Postulate |

# Answer:
The completed proof table has:

1) Statements: $ \overline{DE} \perp \overline{AC} $ and $ E $ is the midpoint of $ \overline{AC} $; Reason: Given

2) Reason: Definition of Midpoint

3) Statements: $ \angle AED \cong \angle CED $; Reason: Definition of Perpendicular Lines (Right angles are congruent)

4) Statements: $ \overline{DE} \cong \overline{DE} $; Reason: Reflexive Property of Congruence

6) Reason: SAS Congruence Postulate

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QUESTION IMAGE

Question

Explanation:

Step1: Identify Given Information

Step2: Midpoint Definition

Step3: Perpendicular Angles

Step4: Common Side

Step5: SAS Congruence

Answer:

statements	reasons
2) \\(\overline{ae} \cong \overline{ce}\\)	2)
3)	3)
4)	4)
5)	5)
6) \\(\delta aed \cong \delta ced\\)	6)

Statements	Reasons
2) \( \overline{AE} \cong \overline{CE} \)	2) Definition of Midpoint
3) \( \angle AED \cong \angle CED \)	3) Definition of Perpendicular Lines (Right angles are congruent)
4) \( \overline{DE} \cong \overline{DE} \)	4) Reflexive Property of Congruence
5) (Not needed as we can go to congruence)	5) (N/A)
6) \( \triangle AED \cong \triangle CED \)	6) SAS Congruence Postulate