初中数学七年级上册（581题）

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如图，\(O\)为直线\(AB\)上一点，\(\angle DOE = 90^\circ \)，\(OD\)是\(\angle AOC\)的角平分线，求证：\(OE\)平分\(\angle BOC\)。

知识点：第四章　几何图形初步

参考答案：解：∵\(OD\)是\(\angle AOC\)的角平分线

∴\(\angle COD = \frac{1}{2}\angle AOC\)

∵\(\angle AOD + \angle BOD = 180^\circ \)

\(\angle BOD = 180^\circ - \angle AOD\)

∵\(\angle DOE = 90^\circ \)

∴\(\angle EOC = \angle EOD - \angle COD = 90^\circ - \frac{1}{2}\angle AOC\)

∴\(\angle EOC = \frac{1}{2}\angle BOC\)

∴\(OE\)平分\(\angle BOC\)