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Cho tứ giác ABCD. Chứng minh rằng tứ giác đó là hình bình hành khi và chỉ khi

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Đề bài

Cho tứ giác ABCD. ♒Chứng minh rằng tứ giác đó là hình bình hành khi và chỉ khi \(\overrightarrow {AB}  = \overrightarrow {DC} \)

Phương pháp giải - Xem chi tiết

\(\overrightarrow {AB}  = \overrightarrow {DC} \) nếu  + \(\overrightarrow {AB}  ,\, \overrightarrow {DC} \) cùng hướng + \(AB =CD\)

Lời giải chi tiết

Tứ giác ABCD là hình bình hành 

\( \Leftrightarrow \left\{ \begin{array}{l}
AB // DC\\
AB = DC
\end{array} \right.\)

Mà \(AB // DC \Leftrightarrow \overrightarrow {AB}  ,\, \overrightarrow {DC} \) cùng phương, do đó cùng hướng.

\( \Leftrightarrow \left\{ \begin{array}{l}
\overrightarrow {AB} , \overrightarrow {DC} \,{\rm{ cùng hướng}}\\
AB = DC
\end{array} \right.\)

\(\Leftrightarrow \overrightarrow {AB}  = \overrightarrow {DC} \)Vậy tứ giác ABCD là hình bình hành khi và chỉ khi \(\overrightarrow {AB}  = \overrightarrow {DC} \).

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