1. (2024·太仓期末)如图，在$△ ABC$中，$AC=BC$，$∠ B=40^{\circ}$，$D$是边$AB$上一点，点$B$关于直线$CD$的对称点为$B'$，当$B'D// AC$，则$∠ BCD$的度数为(
A.$25^{\circ}$
B.$30^{\circ}$
C.$35^{\circ}$
D.$40^{\circ}$
B
)A.$25^{\circ}$
B.$30^{\circ}$
C.$35^{\circ}$
D.$40^{\circ}$
答案：1. B
2. (2024·江阴月考)如图，在$△ ABC$中，$∠ ACB=90^{\circ}$，$∠ B-∠ A=10^{\circ}$，$D$是$AB$上一点，$△ ACD$与$△ ECD$关于$CD$对称，边$CE$交$AB$于点$F$.若$△ DEF$是直角三角形，则$∠ ACD=$
$25^{\circ}$或$5^{\circ}$
.答案：2. $25^{\circ}$或$5^{\circ}$ 点拨:在$△ ABC$中，$∠ ACB = 90^{\circ}$，$∠ B - ∠ A = 10^{\circ}$，$\therefore∠ A = 40^{\circ}$，$∠ B = 50^{\circ}$。
由对称的性质得$∠ E = ∠ A = 40^{\circ}$，$∠ ACD = ∠ ECD$。
当$∠ DFE = 90^{\circ}$时，则$∠ CFB = 90^{\circ}$，
$\therefore∠ BCF = 90^{\circ} - ∠ B = 40^{\circ}$，
$\therefore∠ ACE = ∠ ACB - ∠ BCF = 50^{\circ}$，
$\therefore∠ ACD = \frac{1}{2}∠ ACE = 25^{\circ}$；
当$∠ EDF = 90^{\circ}$时，
$\because∠ E = 40^{\circ}$，$\therefore∠ CFB = ∠ DFE = 50^{\circ}$，
$\therefore∠ BCF = 180^{\circ} - ∠ CFB - ∠ B = 80^{\circ}$，
$\therefore∠ ACE = ∠ ACB - ∠ BCF = 10^{\circ}$，
$\therefore∠ ACD = \frac{1}{2}∠ ACE = 5^{\circ}$。综上所述，$∠ ACD$度数为$25^{\circ}$或$5^{\circ}$。
由对称的性质得$∠ E = ∠ A = 40^{\circ}$，$∠ ACD = ∠ ECD$。
当$∠ DFE = 90^{\circ}$时，则$∠ CFB = 90^{\circ}$，
$\therefore∠ BCF = 90^{\circ} - ∠ B = 40^{\circ}$，
$\therefore∠ ACE = ∠ ACB - ∠ BCF = 50^{\circ}$，
$\therefore∠ ACD = \frac{1}{2}∠ ACE = 25^{\circ}$；
当$∠ EDF = 90^{\circ}$时，
$\because∠ E = 40^{\circ}$，$\therefore∠ CFB = ∠ DFE = 50^{\circ}$，
$\therefore∠ BCF = 180^{\circ} - ∠ CFB - ∠ B = 80^{\circ}$，
$\therefore∠ ACE = ∠ ACB - ∠ BCF = 10^{\circ}$，
$\therefore∠ ACD = \frac{1}{2}∠ ACE = 5^{\circ}$。综上所述，$∠ ACD$度数为$25^{\circ}$或$5^{\circ}$。
3. 如图，直线$l$分别与直线$AB$，$CD$相交于点$E$，$F$，$AB// CD$，$P$是射线$EA$上的一个动点，点$P$，$E$不重合，连接$PF$.点$N$与点$E$关于直线$PF$对称.
(1)当直线$l$与直线$AB$所夹的角$β =50^{\circ}$时.
①若$∠ PFD$的平分线是$FE$，请求出$∠ PFE$的度数；
②若点$N$恰好落在直线$CD$上，试求出$∠ PFE$的度数.
(2)当$∠ CFN=\dfrac{1}{3}∠ CFP=\dfrac{1}{3}β$时，试求出$∠ PFE$的度数.
(1)当直线$l$与直线$AB$所夹的角$β =50^{\circ}$时.
①若$∠ PFD$的平分线是$FE$，请求出$∠ PFE$的度数；
②若点$N$恰好落在直线$CD$上，试求出$∠ PFE$的度数.
(2)当$∠ CFN=\dfrac{1}{3}∠ CFP=\dfrac{1}{3}β$时，试求出$∠ PFE$的度数.
答案：
3. 解:(1)①$\because AB// CD$，$β = 50^{\circ}$，$\therefore∠ EFD = β = 50^{\circ}$。
$\because FE$平分$∠ PFD$，$\therefore∠ PFE = ∠ EFD = 50^{\circ}$。
②当点$N$落在$AB$上时，连接$PN$，如答图①。
$\because$点$N$与点$E$关于直线$PF$对称，直线$PF$是对称轴，$\therefore∠ PFN = ∠ PFE$。
$\becauseβ = 50^{\circ}$，$\therefore∠ PEF = β = 50^{\circ}$。
$\because AB// CD$，$\therefore∠ PEF + ∠ CFE = 180^{\circ}$，
$\therefore∠ CFE = 180^{\circ} - ∠ PEF = 130^{\circ}$，
$\therefore∠ PFE = \frac{1}{2}∠ CFE = 65^{\circ}$。
(2)设$∠ CFN = x$，
$\because∠ CFN = \frac{1}{3}∠ CFP = \frac{1}{3}β$，$\therefore∠ CFP = β = 3x$。
①当点$N$在平行线$AB$，$CD$之间时，如答图②。
$\because∠ CFP = β = 3x$，$∠ CFN = x$，$AB// CD$，
$\therefore∠ PFN = ∠ CFP - ∠ CFN = 2x$，$∠ EFD = β = 3x$。
由对称可得$∠ PFE = ∠ PFN = 2x$。
$\because AB// CD$，
$\therefore∠ AEF + ∠ CFE = 3x + 2x + 2x + x = 180^{\circ}$，
$\therefore x = 22.5^{\circ}$，$\therefore∠ PFE = 2x = 45^{\circ}$；
②当点$N$在$CD$的下方时，$∠ PFN = ∠ CFP + ∠ CFN = 4x$，
由对称可得$∠ PFE = ∠ PFN = 4x$，
$\because AB// CD$，$\therefore∠ AEF + ∠ CFE = 3x + 4x + 3x = 180^{\circ}$，
$\therefore x = 18^{\circ}$，$∠ PFE = 4x = 72^{\circ}$。
综上所述，$∠ PFE$的度数是$45^{\circ}$或$72^{\circ}$。
3. 解:(1)①$\because AB// CD$，$β = 50^{\circ}$，$\therefore∠ EFD = β = 50^{\circ}$。
$\because FE$平分$∠ PFD$，$\therefore∠ PFE = ∠ EFD = 50^{\circ}$。
②当点$N$落在$AB$上时，连接$PN$，如答图①。
$\because$点$N$与点$E$关于直线$PF$对称，直线$PF$是对称轴，$\therefore∠ PFN = ∠ PFE$。
$\becauseβ = 50^{\circ}$，$\therefore∠ PEF = β = 50^{\circ}$。
$\because AB// CD$，$\therefore∠ PEF + ∠ CFE = 180^{\circ}$，
$\therefore∠ CFE = 180^{\circ} - ∠ PEF = 130^{\circ}$，
$\therefore∠ PFE = \frac{1}{2}∠ CFE = 65^{\circ}$。
(2)设$∠ CFN = x$，
$\because∠ CFN = \frac{1}{3}∠ CFP = \frac{1}{3}β$，$\therefore∠ CFP = β = 3x$。
①当点$N$在平行线$AB$，$CD$之间时，如答图②。
$\because∠ CFP = β = 3x$，$∠ CFN = x$，$AB// CD$，
$\therefore∠ PFN = ∠ CFP - ∠ CFN = 2x$，$∠ EFD = β = 3x$。
由对称可得$∠ PFE = ∠ PFN = 2x$。
$\because AB// CD$，
$\therefore∠ AEF + ∠ CFE = 3x + 2x + 2x + x = 180^{\circ}$，
$\therefore x = 22.5^{\circ}$，$\therefore∠ PFE = 2x = 45^{\circ}$；
②当点$N$在$CD$的下方时，$∠ PFN = ∠ CFP + ∠ CFN = 4x$，
由对称可得$∠ PFE = ∠ PFN = 4x$，
$\because AB// CD$，$\therefore∠ AEF + ∠ CFE = 3x + 4x + 3x = 180^{\circ}$，
$\therefore x = 18^{\circ}$，$∠ PFE = 4x = 72^{\circ}$。
综上所述，$∠ PFE$的度数是$45^{\circ}$或$72^{\circ}$。