初中数学八年级上册试题库（555题）

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如图：在\(\Delta ABC\)中，\(BE\)、\(CF\)分别是\(AC\)、\(AB\)两边上的高，在\(BE\)上截取\(BD = AC\)，在

\(CF\)的延长线上截取\(CG = AB\)，连接\(AD\)、\(AG\)。观察线段\(AD\)与\(AG\)有什么关系，并说明

理由。

知识点：第十二章　全等三角形

参考答案：线段\(AD\)与\(AG\)垂直且相等\(\because BE \bot AC\)，\(CF \bot AB\)，\(\therefore \angle HFB = \angle HEC = 90^\circ \)，又\(\because \angle BHF = \angle CHE\)，\(\therefore \angle ABD = \angle ACG\)，在\(\Delta ABD\)和\(\Delta GCA\)中\(\left\{ {\begin{array}{*{20}{l}}
{AB = CG} \\
{\angle ABD = \angle ACG} \\
{BD = CA}
\end{array}} \right.\)，\(\therefore \Delta ABD \cong \Delta GCA(SAS)\)，\(\therefore AD = GA\)（全等三角形的对应边相等）；\(\because \Delta ABD \cong \Delta GCA\)，\(\therefore \angle ADB = \angle GAC\)，又\(\because \angle ADB = \angle AED + \angle DAE\)，\(\angle GAC = \angle GAD + \angle DAE\)，\(\therefore \angle AED = \angle GAD = 90^\circ \)，\(\therefore AD \bot GA\)。