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Giải bài 5 trang 92 SGK Toán 7 tập 2 - Cánh diều

Cho tam giác ABC có

🦋Tổng hợp đề thi học kì 2 lớp 7 tất cả các môn - Cánh diều

Toán - Văn - Anh - Khoa học tự nhiên...
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Đề bài

Cho tam giác ABC có \(\widehat B > \widehat C\). Tia phân giác góc BAC cắt cạnh BC tại điểm D.

a) Chứng minh \(\widehat {ADB} < \widehat {ADC}\).

b) Kẻ tia Dx nằm trong góc ADC sao cho \(\widehat {ADx} = \widehat {ADB}\). Giả sử tia Dx cắt cạnh AC tại điểm Eꦓ. Chứng minh: \(\Delta ABD = \Delta AED,AB < AC\). 

Phương pháp giải - Xem chi tiết

a) Tổng ba góc trong một tam giác bằng 180°.

b) Chứng minh \(\Delta ABD = \Delta AED\) theo trường hợp g.c.g và AB < AC vì cạnh đối diện với góc lớn hơn thì lớn hơn.

Lời giải chi tiết

a) Ta có: \(\widehat {BAD} = \widehat {CAD}\) (vì AD là phân giác của góc BAC).

Mà \(\widehat B > \widehat C\) nên \(\widehat B + \widehat {BAD} > \widehat C + \widehat {CAD}\). Tổng ba góc trong một tam giác bằng 180° nên: \(\begin{array}{l}\widehat B + \widehat {BAD} > \widehat C + \widehat {CAD}\\ 180^\circ  - (\widehat B + \widehat {BAD}) < 180^\circ  - (\widehat C + \widehat {CAD})\\  \widehat {ADB} < \widehat {ADC}\end{array}\)

b) Xét hai tam giác ADB và tam giác ADE có:

     \(\widehat {ADB} = \widehat {ADE}\);

     AD chung;

     \(\widehat {BAD} = \widehat {EAD}\). Vậy \(\Delta ABD = \Delta AED\) (g.c.g) Trong một tam giác, cạnh đối diện với góc lớn hơn thì lớn hơn.

Trong tam giác ABC có \(\widehat B > \widehat C\) nên AC > AB hay AB < AC (AB là cạnh đối diện với góc C, AC là cạnh đối diện với góc B).

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