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Bài tập 17 trang 116 Tài liệu dạy – học Toán 7 tập 1

Giải bài tập Cho góc tù xOy. Trong góc xOy vẽ tia Om vuông góc với Ox và tia On vuông góc với tia Oy.
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Đề bài

Cho góc tù xOy. Trong góc xOy vẽ tia Om vuông góc với Ox và tia On vuông góc với tia Oy. a) Chứng tỏ rằng \(\widehat {xOn} = \widehat {mOy}\) b) Tính số đo góc \(\widehat {xOy} + \widehat {mOn}\) c) Gọi Ot là tia phân giác của góc xOy. Chứng minh rằng Ot là tia phân giác của góc mOn.

Lời giải chi tiết

 

a)Ta có: \(Om \bot 0x \Rightarrow \widehat {m0x} = {90^0}\)  và \(On \bot Oy \Rightarrow \widehat {nOy} = {90^0}\)Do đó: \(\widehat {m0x} = \widehat {nOy}({90^0})(1)\)Tia On nằm trong \(\widehat {xOy} \Rightarrow \widehat {xOn} + \widehat {nOy} = \widehat {xOy}\)Và tia Om nằm trong \(\widehat {xOy} \Rightarrow \widehat {mOy} + \widehat {m0x} = \widehat {xOy}\)Nên \(\widehat {xOn} + \widehat {nOy} = \widehat {mOy} + \widehat {m0x}(2)\)Từ (1) và (2) ta có: \(\widehat {xOn} = \widehat {mOy}\)b) Ta có: \(\eqalign{  & \widehat {xOy} + \widehat {mOn} = \widehat {nOy} + \widehat {xOn} + \widehat {mOn}  \cr  &  = {90^0} + \widehat {mOy} + \widehat {mOn} = {90^0} + \widehat {nOy} = {90^0} + {90^0} = {180^0} \cr} \)c) Ta có: Ot là tia phân giác góc \(\widehat {xOy} \Rightarrow \widehat {xOt} = \widehat {yOt} \Rightarrow \widehat {xOn} + \widehat {nOt} = \widehat {yOm} + \widehat {mOt}\)Mà \(\widehat {xOn} = \widehat {mOy}\)  (câu a) \( \Rightarrow \widehat {nOt} = \widehat {mOt}\)Mà tia Ot nằm giữa hai tia Om và On => Ot là tia phân giác góc mOn.

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