初中数学八年级上册试题库（555题）

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如图，已知\(AC\)平分\(\angle BAD\)，\(CE \bot AB\)于\(E\)，\(CF \bot AD\)于\(F\)，且\(BC = CD\)．求证：

（1）\(\Delta BCE \cong \Delta DCF\)。

（2）\(AB + AD = 2AE\)。

知识点：第十二章　全等三角形

参考答案：证明：（1）\(\because AC\)是角平分线，\(CE \bot AB\)于\(E\)，\(CF \bot AD\)于\(F\)，\(\therefore CE = CF\)，在\({\rm{Rt}}\Delta {\rm{BCE}}\)和\({\rm{Rt}}\Delta {\rm{DCF}}\)中，\(\left\{ {\begin{array}{*{20}{l}}
{BC = DC} \\
{CE = CF}
\end{array}} \right.\)\(\therefore \Delta BCE \cong \Delta DCF(HL)\)；（2）在\({\rm{Rt}}\Delta {\rm{FAC}}\)和\({\rm{Rt}}\Delta {\rm{EAC}}\)中，\(\left\{ {\begin{array}{*{20}{l}}
{AC = AC} \\
{CE = CF}
\end{array}} \right.\)，\(\therefore Rt\Delta FAC \cong Rt\Delta EAC(HL)\)，\(\therefore AF = AE\)，\(\because \Delta BCE \cong \Delta DCF\)，\(\therefore BE = DF\)，\(\therefore AB + AD = (AE + BE) + (AF - DF) = 2AE\)。