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1. 在$Rt△ABC$中,$∠B= 2∠C$,则$∠C$的度数为______.
答案
$45^{\circ}$或$30^{\circ}$
解:①当$\angle B = 90^{\circ}$时,$2\angle C = 90^{\circ}$,
$\therefore \angle C = 45^{\circ}$;
②当$\angle A = 90^{\circ}$时,$\angle B + \angle C = 90^{\circ}$,$\therefore 2\angle C + \angle C = 90^{\circ}$,
$\therefore \angle C = 30^{\circ}$.
解:①当$\angle B = 90^{\circ}$时,$2\angle C = 90^{\circ}$,
$\therefore \angle C = 45^{\circ}$;
②当$\angle A = 90^{\circ}$时,$\angle B + \angle C = 90^{\circ}$,$\therefore 2\angle C + \angle C = 90^{\circ}$,
$\therefore \angle C = 30^{\circ}$.
2. 如图,在$△ABC$中,$∠A= 50^{\circ },∠B= 30^{\circ }$,点 D 在 AB 边上,连接 CD,若$△ACD$为直角三角形,则$∠BCD$的度数为______.
答案
$60^{\circ}$或$10^{\circ}$
解:①当$\angle ADC = 90^{\circ}$时,
$\angle BCD = 90^{\circ} - 30^{\circ} = 60^{\circ}$;
②当$\angle ACD = 90^{\circ}$时,
$\angle ACB = 180^{\circ} - 30^{\circ} - 50^{\circ} = 100^{\circ}$,
$\angle BCD = 100^{\circ} - 90^{\circ} = 10^{\circ}$.
3. 在$△ABC$中,AD 是 BC 边上的高,$∠BAD= 60^{\circ },∠CAD= 20^{\circ }$,AE 为$△ABC$的角平分线,则$∠EAD$的度数为______.
答案
$20^{\circ}$或$40^{\circ}$
解:①当高$AD$在$\triangle ABC$内部时,$\angle EAD = 20^{\circ}$;
②当高$AD$在$\triangle ABC$外部时,
$\angle EAD = 40^{\circ}$.
4. 在$△ABC$中,$∠ABC= ∠C$,BD 是 AC 边上的高,$∠ABD= 40^{\circ }$,则$∠C$的度数为______.
答案
$25^{\circ}$或$65^{\circ}$
解:①$\angle BAC$为锐角时,$\angle C = 65^{\circ}$;②$\angle BAC$为钝角时,$\angle C = 25^{\circ}$.
5. 在$△ABC$中,$∠A= 50^{\circ }$,BD,CE 是它的两条高,直线 BD,CE 交于点 O,则$∠BOC$的度数为______.
答案
$130^{\circ}$或$50^{\circ}$
解:①当$\triangle ABC$为锐角三角形时,$\angle BOC = 130^{\circ}$;
②当$\triangle ABC$为钝角三角形时，
$\angle BOC = 50^{\circ}$.
