Given: start overline, B, D, end overline, \cong, start overline, B, E, end overline, comma 
BD
 ≅ 
BE
 , start overline, D, E, end overline 
DE
  bisects start overline, A, B, end overline 
AB
  and start overline, D, E, end overline 
DE
  bisects start overline, B, C, end overline, . 
BC
 .

Prove: start overline, A, B, end overline, \cong, start overline, B, C, end overline 
AB
 ≅ 
BC
 .


We are given the following information:

1. \( \overline{BD} \cong \overline{BE} \) (i.e., BD is congruent to BE),
2. \( \overline{DE} \) bisects \( \overline{AB} \) and \( \overline{BC} \).

We are asked to prove that \( \overline{AB} \cong \overline{BC} \) (i.e., AB is congruent to BC).

### Proof:

1. **Given**: \( \overline{BD} \cong \overline{BE} \). Since \( D \) and \( E \) are on opposite sides of \( B \), it implies that \( B \) is the midpoint of \( \overline{DE} \).
   
   Thus, \( \triangle BDE \) is isosceles, and \( \overline{BD} \cong \overline{BE} \).

2. **Given**: \( \overline{DE} \) bisects \( \overline{AB} \) and \( \overline{BC} \). This means:
   \[
   AE \cong EB \quad \text{(since DE bisects AB)}
   \]
   \[
   BE \cong EC \quad \text{(since DE bisects BC)}
   \]
   Therefore, point \( E \) is the midpoint of \( \overline{AB} \) and \( \overline{BC} \).

3. Now, observe that since \( E \) is the midpoint of both \( \overline{AB} \) and \( \overline{BC} \), we have:
   \[
   AB = BE = BC
   \]

4. From this, it directly follows that:
   \[
   \overline{AB} \cong \overline{BC}
   \]

Thus, we have proven that \( \overline{AB} \cong \overline{BC} \).

### Conclusion:
The congruence of \( \overline{AB} \) and \( \overline{BC} \) follows from the fact that \( E \), being the midpoint of both segments, ensures both sides are equal in length.

Would you like more details on any part of the proof, or do you have any questions?

---

Here are 5 related questions to expand on this concept:

1. What are some properties of isosceles triangles?
2. How does the concept of a midpoint relate to segment bisectors?
3. Can we always say that if two sides are congruent, the triangles they form are congruent?
4. How do you prove two triangles are congruent using the Side-Angle-Side (SAS) theorem?
5. What is the significance of the bisector of an angle or segment in geometric proofs?

### Tip:
When trying to prove congruence of segments or angles, look for symmetry or midpoints, as they often indicate equal lengths or angles.

Given: BD ≅ BE, DE bisects AB and DE bisects BC. Prove: AB ≅ BC.

Prove that AB ≅ BC given BD ≅ BE, and DE bisects both AB and BC. This geometry proof involves concepts of congruence, midpoints, isosceles triangles, and segment bisectors, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation