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Giải bài tập 5 trang 79 SGK Toán 9 tập 2 - Cánh diều

Cho tứ giác ABCD và các điểm M, N lần lượt thuộc các đoạn thẳng AB và CD sao cho các tứ giác AMND, BMNC là các tứ giác nội tiếp. Chứng minh (widehat A + widehat B = 180^circ .)

Tổng hợp Đề thi vào 10 có đáp án và lời giải

Toán - Văn - Anh
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Đề bài

Cho tứ giác ABCD và các điểm M, N lần lượt thuộc các đoạn thẳng AB và CD  sao cho các tứ giác AMND, BMNC là các tứ giác nội tiếp. Chứng minh \(\widehat A + \widehat B = 180^\circ .\)

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

Áp dụng tính chất: Hình chữ nhật có 2 đường chéo bằng nhau và cắt nhau tại trung điểm của mỗi đường để chứng minh AO = CO = OB = OD = R.

Lời giải chi tiết

Do AMND nội tiếp nên \(\widehat A + \widehat {MND} = 180^\circ \) (1) và MBCN nội tiếp nên \(\widehat B + \widehat {MNC} = 180^\circ \left( 2 \right).\) Ta lại có \(\widehat {MND} + \widehat {MNC} = 180^\circ \)(kề bù) (3) Cộng vế với vế của (1), (2), và kết hợp với (3) ta có:

 \(\begin{array}{l}\widehat A + \widehat {MND} + \widehat {MNC} + \widehat B = 180^\circ  + 180^\circ  = 360^\circ \\\widehat A + \widehat B = 360^\circ  - \left( {\widehat {MND} + \widehat {MNC}} \right)\\\widehat A + \widehat B = 360^\circ  - 180^\circ  = 180^\circ \end{array}\)

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