Question

Write a proof for the following \overline{AB}\cong \overline{DE} and C is the midpoint of both \overline{AB}\ and\ \overline{BE}

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ANSWERED
asked 2021-08-09
Write a proof for the following:
Given \(\displaystyle\overline{{{A}{B}}}\stackrel{\sim}{=}\overline{{{D}{E}}}\) and C is the midpoint of both \(\displaystyle\overline{{{A}{B}}}\ {\quad\text{and}\quad}\ \overline{{{B}{E}}}\).
Prove: \(\displaystyle\triangle{A}{B}{C}\stackrel{\sim}{=}\triangle{D}{E}{C}\).
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Answers (1)

2021-08-10
Step 1
\(\displaystyle{A}{B}\stackrel{\sim}{=}{D}{E}\) and C is the mid point of both AD and BE.
Now
AC=DC...(1) (\(\displaystyle\because\) C is mid point of AD)
CE=BC...(2) (\(\displaystyle\because\) C is the mid point of BE)
\(\displaystyle\angle{A}{C}{B}=\angle{B}{C}{E}\)...(3) (\(\displaystyle\because\) vertically opposite \(\displaystyle\angle{s}\))
Also \(\displaystyle{A}{B}\stackrel{\sim}{=}{D}{E}\)...(4) (Given)
Step 2
Therefore from (1), (2) and (4)
\(\displaystyle\triangle{A}{B}{C}\stackrel{\sim}{=}\triangle{D}{E}{C}\) (by SSS similarity)
Also from (1), (2) and (4)
\(\displaystyle\triangle{A}{B}{C}\stackrel{\sim}{=}\triangle{D}{E}{C}\) (by SAS similarity)
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