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Bài tập 5 trang 96 Tài liệu dạy – học Toán 7 tập 2

Giải bài tập Cho tam giác ABC nhọn (AB < AC)
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Đề bài

Cho tam giác ABC nhọn (AB < AC). Tia phân giác của \(\widehat {BAC}\) cắt BC ở D. Trên tia AC lấy điểm E sao cho AE = AB. a) Chứng minh: \(\widehat {DEC} > \widehat {ADB}\) b) So sánh độ dài BD và DC.

Lời giải chi tiết

 

a) Xét ∆ADE và ∆ADB ta có:AE = AB (gt)\(\widehat {DAE} = \widehat {BAD}\) (AD là tia phân giác của \(\widehat {BAC}\))AD (cạnh chung)Do đó ∆ADE = ∆ADB (c.g.c) \( \Rightarrow \widehat {ADE} = \widehat {ADB}\)Mà \(\widehat {DEC}\) là góc ngoài của tam giác ADENên \(\widehat {DEC} > \widehat {ADE} \Rightarrow \widehat {DEC} > \widehat {ADB}.\)b) Ta có \(\widehat {ADB} > \widehat {DCE}(\widehat {ADB}\) là góc ngoài của tam giác ACD)Mà \(\widehat {DEC} > \widehat {ADB}\) (câu a) \( \Rightarrow \widehat {DEC} > \widehat {DCE}\)∆CDE có \(\widehat {DEC} > \widehat {DCE} \Rightarrow\) DC > ED (định lí cạnh đối diện với góc lớn hơn)Mà ED = BD (∆ADE = ∆ADB). Do vậy DC>BD.

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