given: $ \overline { a b } \cong \overline { b c } $, $ \overline { d e } $ bisects $ \overline { a b } $ and $ \overline { d e } $ bisects $ \overline { b c } $.
prove: $ \overline { b d } \cong \overline { b e } $.
note: quadrilateral properties are not permitted in this proof.
1 $ \overline { a b } \cong \overline { b c } $ $ \overline { d e } $ bisects $ \overline { a b } $ $ \overline { d e } $ bisects $ \overline { b c } $
2 $ b d = \frac { 1 } { 2 } b a $
3 $ b e = \frac { 1 } { 2 } b c $

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# Explanation:
## Step1: Use the definition of a bisector
Since $ \overline{DE} $ bisects $ \overline{AB} $, by the definition of a bisector, $ BD=\frac{1}{2}BA $.
Since $ \overline{DE} $ bisects $ \overline{BC} $, by the definition of a bisector, $ BE = \frac{1}{2}BC $.
## Step2: Use the property of congruent segments
Given $ \overline{AB}\cong\overline{BC} $, then $ AB = BC $ (congruent segments have equal lengths).
## Step3: Substitute $ AB$ with $ BC$
Substitute $ AB$ with $ BC$ in the equation $ BD=\frac{1}{2}BA $. We get $ BD=\frac{1}{2}BC $.
Since $ BE=\frac{1}{2}BC $ and $ BD=\frac{1}{2}BC $, then $ BD = BE $.
## Step4: Use the definition of congruent segments
Since $ BD = BE $, by the definition of congruent segments ($ \overline{XY}\cong\overline{ZW} $ if $ XY = ZW $), we have $ \overline{BD}\cong\overline{BE} $.

# Answer:
$ \overline{BD}\cong\overline{BE} $ is proved.

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QUESTION IMAGE

Question

Explanation:

Step1: Use the definition of a bisector

Step2: Use the property of congruent segments

Step3: Substitute \( AB\) with \( BC\)

Step4: Use the definition of congruent segments

Answer: