8 如图,在△ABC 和△EDF 中,∠B=∠D=90°,点A,E,C,F 在同一条直线上,AE=CF,BC 的延长线交DF 于点M,∠MCF=∠F.求证:BC=DF.

8. $\because ∠ MCF = ∠ F, ∠ MCF = ∠ ACB, \therefore ∠ ACB = ∠ F$.
$\because AE=CF,\therefore AE+EC=CF+EC$,即$AC=EF$.在$△ ABC$ 和$△ EDF$ 中,$\begin{cases}∠ B=∠ D,\\∠ ACB=∠ F,\\AC=EF,\end{cases}$ $\therefore △ ABC ≌ △ EDF (\mathrm{AAS}),\therefore BC=DF$

9 [2025 内江改编]如图，点A,B,C,D在同一条直线上，点E,F分别在直线AB的两侧，且$AE=BF$，$AE// BF$，$∠ ECD=∠ FDC$.
(1) 求证:$CE=DF$；
(2) 若$AC=2$,$CD=4$,求$AB$的长.

9. (1) $\because AE//BF,\therefore ∠ A=∠ B. \because ∠ ACE+∠ ECD=180°$,$∠ BDF + ∠ FDC = 180°$, $∠ ECD = ∠ FDC, \therefore ∠ ACE = ∠ BDF$. 在 $△ ACE$ 和 $△ BDF$ 中, $\begin{cases}∠ A=∠ B,\\∠ ACE=∠ BDF,\\AE=BF,\end{cases}$ $\therefore △ ACE≌△ BDF (\mathrm{AAS}), \therefore CE = DF$
(2) 由(1)知，$△ ACE≌△ BDF,\therefore AC=BD=2. \because CD=4,\therefore AB=AC+CD+BD=2+4+2=8,\therefore AB$ 的长为8

10 如图,点A,F,E,C在同一条直线上,AB//CD,∠ABE=∠CDF,AF+AE=AC.
(1) ① $△ ABE ≌$ ;
② $△ BCE ≌$ ;
③ $△ ABC ≌$ .
(2) 对(1)中的①②加以证明.

10. (1) ① $△ CDF$ ② $△ DAF$ ③ $△ CDA$
(2) ① $\because AB//CD,\therefore ∠ BAE=∠ DCF. \because AF+AE=AC,\therefore AE=AC-AF = CF$. 在$△ ABE$ 和 $△ CDF$ 中, $\begin{cases}∠ ABE=∠ CDF,\\∠ BAE=∠ DCF,\\AE=CF,\end{cases}$ $\therefore △ ABE ≌ △ CDF (\mathrm{AAS})$.
② $\because △ ABE ≌ △ CDF,\therefore ∠ AEB=∠ CFD,BE=DF,\therefore ∠ BEC=∠ DFA. \because AF+AE=AC,CE+AE=AC,\therefore AF=CE$.在$△ BCE$和$△ DAF$中,$\begin{cases}BE=DF,\\∠ BEC=∠ DFA,\\CE=AF,\end{cases}$ $\therefore △ BCE≌△ DAF(\mathrm{SAS})$