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Bài tập 6 trang 156 Tài liệu dạy – học Toán 7 tập 1

Giải bài tập Cho tam giác DEF vuông tại D, phân giác của góc E cắt DF tại A. Trên EF lấy điểm B sao cho EB = ED.
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Đề bài

Cho tam giác DEF vuông tại D, phân giác của góc E cắt DF tại A. Trên EF lấy điểm B sao cho EB = ED. a) Chứng minh rằng \(\Delta DEA = \Delta BEA\) b) Chứng minh rằng \(AB \bot EF\)

Lời giải chi tiết

 

a)Xét tam giác DEA và BEA có:ED = EB (gt)\(\widehat {DEA} = \widehat {BEA}\)   (EA là tia phân giác của góc DEB)EA là cạnh chung.Do đó: \(\Delta DEA = \Delta BEA(c.g.c)\)b) Ta có: \(\Delta DEA = \Delta BEA \Rightarrow \widehat {DAE} = \widehat {BAE}\)Tam giác ADE vuông tại D có: \(\widehat {DEA} + \widehat {DAE} = {90^0}\)Mà  \(\widehat {DEA} = \widehat {AEB}\)  (EA là tia phân giác của góc DEB) và  \(\widehat {DAE} = \widehat {BAE}\)Nên  \(\widehat {DEA} + \widehat {DAE} = {90^0} \Leftrightarrow \widehat {AEB} + \widehat {BAE} = {90^0}.\)Mặt khác: \(\widehat {ABF} = \widehat {AEB} + \widehat {BAE}\)  (góc ngoài của tam giác ABE)Do đó: \(\widehat {ABF} = {90^0} \Rightarrow AB \bot EF\)

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