题干

如图,在▱ABCD中,BD是对角线,AE⊥BD于点E,CF⊥BD于点F,试判断四边形AECF是不是平行四边形,并说明理由.

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答案(点此获取答案解析)

解:四边形AECF是平行四边形.

理由如下:

∵AE⊥BD于点E,CF⊥BD于点F,

∴∠AEF=∠CFE=90°,

∴AE∥CF(内错角相等,两直线平行),

在平行四边形ABCD中,AB=CD,AB∥CD,

∴∠ABE=∠CDF,

在△ABE与△DCF中, {

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    吕元膺,字景夫,郓州东平人。姿仪瑰秀,有器识。历右司员外郎。出为蕲州刺史,尝录囚,囚或白:“父母在,明日岁旦不得省,为恨。”因泣,元膺恻然,悉释械归之,而戒还期。吏白“不可”,答曰:“吾以信待人,人岂我违?”如期而至。自是群盗感愧,悉避境去。

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(节选自《新唐书·吕元膺传》)