25. (10 分)已知: 在$△ ABC$中, 点$D$在直线$AB$上, 连接$CD$.
(1)如图 1, 若$∠ ACB = 90°$, $CD ⊥ AB$. 求证: $∠ ACD = ∠ ABC$;
(2)若$∠ ACD = ∠ ABC$, $∠ BAC$的平分线与$CB$, $CD$分别交于点$E$, $F$.
①如图 2, 当点$D$在边$AB$上(不与点$A$, 点$B$重合)时, 求证: $∠ CFE = ∠ CEF$;
②当点$D$在$AB$的延长线上时, "$∠ CFE = ∠ CEF$"是否依然成立? 画出图形,并说明理由.

25. 【点拨】本题考查三角形内角和定理、三角形外角的性质、角平分线的定义、垂直的定义.
【解析】(1)证明:$\because CD ⊥ AB, \therefore ∠ CDB = 90°$,
$\therefore ∠ ABC + ∠ BCD = 90°$.
又$\because ∠ ACB = 90°, \therefore ∠ ACD + ∠ BCD = 90°$,
$\therefore ∠ ACD = ∠ ABC$.
(2)①证明:$\because AE$平分$∠ BAC, \therefore ∠ BAE = ∠ CAE$.
$\because ∠ CFE = ∠ CAE + ∠ ACD, ∠ CEF = ∠ BAE + ∠ ABC, ∠ ACD = ∠ ABC$,
$\therefore ∠ CFE = ∠ CEF$.
②如图,当点D在AB的延长线上时,$∠ CFE = ∠ CEF$依然成立. 理由如下:
$\because ∠ ACD = ∠ ABC, ∠ ACD = ∠ ACB + ∠ BCD, ∠ ABC = ∠ D + ∠ BCD$,
$\therefore ∠ ACB = ∠ D$.
$\because ∠ CEF = ∠ CAE + ∠ ACE, ∠ CFE = ∠ DAF + ∠ D, ∠ DAF = ∠ CAE$,
$\therefore ∠ CFE = ∠ CEF$.

26. (10 分)定义:在一个三角形中,如果一个内角α的度数比另一个内角度数大$36°$,那么我们称这样的三角形为“似黄金三角形”,称α为“黄金角”.例如:一个三角形三个内角的度数分别是$30°$,$84°,66°$,这个三角形就是“似黄金三角形”,其中这个$66°$的角为“黄金角”.
(1)一个“似黄金三角形”的一个内角为$92°$,若“黄金角”为锐角,则这个“黄金角”的度数为;
(2)如图1,在$△ ABC$中,$∠ A = 70°,∠ B = 60°,D$为线段$AB$上一点(点$D$不与点$A,B$重合).若$△ BCD$是“似黄金三角形”,求$∠ BDC$的度数;
(3)如图2,在$△ ABC$中,点$D$在边$AB$上,$DE$平分$∠ ADC$交$AC$于点$E$,过点$E$作$EF// AB$交$CD$于点$F$,且$∠ DEF = ∠ B$.若$△ BCD$和$△ ACD$都是“似黄金三角形”,请直接写出$∠ A$的度数.
·72·

26. 【点拨】本题考查新定义“似黄金三角形”、三角形内角和定理、三角形外角的性质、角平分线的定义、平行线的性质、分类讨论思想.
【解析】(1)设这个三角形的黄金角$x$为锐角,则另一个角为$x-36°$.
根据三角形内角和定理,得$x + x - 36° + 92° = 180°$,
解得$x=62°$,$\therefore$ 这个“黄金角”的度数为$62°$.
故答案为$62°$.
(2)$\because ∠ A=70°, ∠ B=60°, \therefore ∠ ACB=180°-70°-60°=50°$,
$\because ∠ BDC = ∠ A + ∠ ACD, \therefore ∠ BDC ＞ ∠ A$,
$\therefore ∠ BDC ＞ ∠ B ＞ ∠ BCD$.
$\because △ BCD$为“似黄金三角形”,
①若$∠ B$为“黄金角”,则$∠ BCD = 60° - 36° = 24°$,
$\therefore ∠ BDC = 180° - 60° - 24° = 96°$;
②$∠ BCD$最小,不可能为“黄金角”;
③若$∠ BDC$为“黄金角”,则$∠ BCD = ∠ BDC - 36°$或$∠ B = ∠ BDC - 36°$.
当$∠ BCD = ∠ BDC - 36°$时,
$\because ∠ B + ∠ BCD + ∠ BDC = 180°$,
$\therefore 60° + ∠ BDC - 36° + ∠ BDC = 180°$,
$\therefore ∠ BDC = 78°$.
当$∠ B = ∠ BDC - 36°$时,$∠ BDC = 60° + 36° = 96°$.
综上所述,$∠ BDC$的度数为$96°$或$78°$.
(3)$\because EF // AB, \therefore ∠ DEF = ∠ ADE$.
$\because ∠ DEF = ∠ B, \therefore ∠ ADE = ∠ B$.
$\because DE$平分$∠ ADC, \therefore ∠ ADC = 2∠ ADE, \therefore ∠ ADC = 2∠ B$.
$\because ∠ ADC = ∠ B + ∠ BCD, \therefore ∠ B = ∠ BCD$.
设$∠ B = ∠ BCD = x$,
$\because △ BCD$为“似黄金三角形”,
①当$∠ BDC = x + 36°$时,$\because ∠ B + ∠ BCD + ∠ BDC = 180°$,
$\therefore x + x + x + 36° = 180°$,解得$x=48°$,
$\therefore ∠ B = ∠ BCD = 48°, \therefore ∠ ADC = 48° + 48° = 96°$.
$\because △ ACD$是“似黄金三角形”,
当$∠ ACD$为“黄金角”,即$∠ ACD = ∠ A + 36°$时,
$\because ∠ A + ∠ ACD + ∠ ADC = 180°$,
$\therefore ∠ A + ∠ A + 36° + 96° = 180°$,
$\therefore ∠ A = 24°$;
当$∠ ADC$为“黄金角”时,则$∠ ADC = ∠ A + 36°$或$∠ ADC = ∠ ACD + 36°$,即$96° = ∠ A + 36°$或$96° = 180° - 96° - ∠ A + 36°$,
$\therefore ∠ A = 60°$或$∠ A = 24°$;
当$∠ A$为“黄金角”时,则$∠ A = ∠ ACD + 36°$,即$∠ A = 180° - 96° - ∠ A + 36°, \therefore ∠ A = 60°$;
②当$∠ BDC = x - 36°$时,
$\because ∠ B + ∠ BCD + ∠ BDC = 180°$,
$\therefore x + x + x - 36° = 180°, \therefore x=72°$,
$\therefore ∠ ADC = 72° + 72° = 144°$,
这种情况下,$△ ACD$不可能是“似黄金三角形”.
综上所述,$∠ A$的度数为$24°$或$60°$.