13. (2024秋·西山区期末)如图,点A,C,E,F在同一条直线上,CD=AB,∠C=∠A,CE=AF.求证:△CDF≌△ABE.

13. $\because CE=AF, \therefore CF=AE$,在$△ CDF$和$△ ABE$中,
$\begin{cases}CD=AB,\\∠ C=∠ A,\\CF=AE,\end{cases}$
$\therefore △ CDF ≌ △ ABE (\mathrm{SAS}).$

14. (2023·苏州)如图,在△ABC中,AB=AC,AD为△ABC的角平分线.以点A圆心,AD长为半径画弧,与AB,AC分别交于点E,F,连接DE,DF.
求证:△ADE≌△ADF.

14. 证明:$\because AD$是$△ ABC$的角平分线,$\therefore ∠ BAD=∠ CAD$. 由作图知:$AE=AF$. 在$△ ADE$和$△ ADF$中,$\begin{cases}AE=AF,\\∠ BAD=∠ CAD,\\AD=AD,\end{cases}$
$\therefore △ ADE ≌ △ ADF (\mathrm{SAS}).$

15. (2024秋·息县期末)如图,点A,D,B,E在一条直线上,AD=BE,AC=DF,AC//DF,求证:△ABC≌△DEF.

15. $\because AD=BE, \therefore AD+BD=BE+BD$,即$AB=DE$. $\because AC// DF, \therefore ∠ A=∠ EDF$,在$△ ABC$与$△ DEF$中,$\begin{cases}AB=DE,\\∠ A=∠ EDF,\\AC=DF,\end{cases}$
$\therefore △ ABC ≌ △ DEF (\mathrm{SAS}).$