Finish the Proof.

Question

Give: Line AC ≅ BC and line Ac and BD bisect
Prove line BE ≅ EC

Step Statement Reason
1
start overline, A, C, end overline, \cong, start overline, B, D, end overline
AC
≅
BD

start overline, A, C, end overline
AC
and start overline, B, D, end overline
BD
bisect each other
Given
2
start overline, A, E, end overline, \cong, start overline, E, C, end overline
AE
≅
EC

A segment bisector divides a segment into two congruent segments
3
start overline, B, E, end overline, \cong, start overline, E, D, end overline
BE
≅
ED

A segment bisector divides a segment into two congruent segments
4
angle, D, E, C, \cong, angle, B, E, A∠DEC≅∠BEA
Vertical angles are congruent
5
triangle, D, E, C, \cong, triangle, B, E, A△DEC≅△BEA
SAS
6
start overline, B, D, end overline, \cong, start overline, B, D, end overline
BD
≅
BD

Reflexive Property
7
start overline, C, A, end overline, \cong, start overline, C, A, end overline
CA
≅
CA

Reflexive Property
8
start overline, A, B, end overline, \cong, start overline, D, C, end overline
AB
≅
DC

Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
9
angle, B, E, C, \cong, angle, A, E, D∠BEC≅∠AED
Vertical angles are congruent

Answer 1

To finish the proof, we will proceed with the statements necessary to conclude that \( \overline{BE} \cong \overline{EC} \). We can integrate our completion into the structure of your proof to show the congruence of these segments. Below are the steps needed to achieve this:

Proof Steps

| Step | Statement | Reason | |------|----------------------------------------|--------| | 1 | \( \overline{AC} \cong \overline{BD} \) | Given | | 2 | \( \overline{AE} \cong \overline{EC} \) | A segment bisector divides a segment into two congruent segments | | 3 | \( \overline{BE} \cong \overline{ED} \) | A segment bisector divides a segment into two congruent segments | | 4 | \( \angle DEC \cong \angle BEA \) | Vertical angles are congruent | | 5 | \( \triangle DEC \cong \triangle BEA \) | SAS (Side-Angle-Side) | | 6 | \( \overline{BD} \cong \overline{BD} \) | Reflexive Property | | 7 | \( \overline{CA} \cong \overline{CA} \) | Reflexive Property | | 8 | \( \overline{AB} \cong \overline{DC} \) | Corresponding Parts of Congruent Triangles are Congruent (CPCTC) | | 9 | \( \angle BEC \cong \angle AED \) | Vertical angles are congruent | | 10 | \( \overline{BE} \cong \overline{EC} \) | CPCTC of triangles \( DEC \) and \( BEA \) |

Explanation of the Final Steps

• Steps 6 and 7 reinforce the reflexive property, demonstrating that the segments on each triangle being compared are congruent to themselves.

• Step 8 uses the congruence of the triangles \( \triangle DEC \) and \( \triangle BEA \) to conclude that their corresponding parts must also be congruent. This is a result of the CPCTC theorem and directly leads to the conclusion that \( \overline{BE} \cong \overline{EC} \).

• The final step provides the conclusion with the required congruence statement that \( \overline{BE} \cong \overline{EC} \), which is what we're trying to prove.

Thus, we have shown that \( \overline{BE} \) is congruent to \( \overline{EC} \), completing the proof.