Math Problem Statement

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 .

Solution

We are given the following information:

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

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

Proof:

  1. Given: BDBE\overline{BD} \cong \overline{BE}. Since DD and EE are on opposite sides of BB, it implies that BB is the midpoint of DE\overline{DE}.

    Thus, BDE\triangle BDE is isosceles, and BDBE\overline{BD} \cong \overline{BE}.

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

  3. Now, observe that since EE is the midpoint of both AB\overline{AB} and BC\overline{BC}, we have: AB=BE=BCAB = BE = BC

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

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

Conclusion:

The congruence of AB\overline{AB} and BC\overline{BC} follows from the fact that EE, 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.

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Math Problem Analysis

Mathematical Concepts

Congruence of Line Segments
Isosceles Triangles
Midpoints
Bisectors

Formulas

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Theorems

Congruence of Segments
Properties of Isosceles Triangles
Midpoint Theorem
Segment Bisector Theorem

Suitable Grade Level

Grades 9-12