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Bài tập 29 trang 98 Tài liệu dạy – học Toán 7 tập 2

Giải bài tập Tam giác ABC cân tại A, trên cạnh BC lấy các điểm D, E sao cho BD = DE = EC. Chứng minh rằng trong ba góc BAD, DAE, EAC thì góc DAE là góc lớn nhất.
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Đề bài

Tam giác ABC cân tại A, trên cạnh BC lấy các điểm D, E sao cho BD = DE = EC. Chứng minh rằng trong ba góc BAD, DAE, EAC thì góc DAE là góc lớn nhất.

Lời giải chi tiết

 

Xét ∆ABD và ∆ACE có: AB = AC (∆ABC cân tại A)\(\widehat {ABD} = \widehat {ACE}\) (∆ABC cân tại A)BD = EC (gt)Do đó ∆ABD = ∆ACE (c.g.c) \( \Rightarrow \widehat {BAD} = \widehat {EAC}\)Ta có \(\widehat {AEB} > \widehat C(\widehat {AEB}\) là góc ngoài của tam giác ACD)\(\widehat C = \widehat B\) (∆ABC cân tại A)Nên \(\widehat {AEB} > \widehat B\)∆ABE có \(\widehat {AEB} > \widehat B\) => AB > AETrên tia đối của tia DA lấy điểm M sao cho DM = DAXét ∆DME và ∆DAB có DM = DA, \(\widehat {MDE} = \widehat {ADB}\) (đối đỉnh), DE = BD (gt)Do đó ∆DME = ∆DAB (c.g.c) \( \Rightarrow ME = AB,\widehat {DME} = \widehat {BAD}\)Ta có ME > AE. ∆AEM có ME > AE \( \Rightarrow \widehat {DAE} > \widehat {DME}\)Nên \(\widehat {DAE} > \widehat {BAD} = \widehat {EAC}.\)Vậy trong ba góc BAD, DAE, EAC thì góc DAE lớn nhất.

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