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八年级数学填空题一般
题目
操作:如图11,将ABC\triangle ABC沿射线BFBF平移到DCE\triangle DCE,使原BB点与CC点重合,这时CDCDABAB,所以1=A\angle 1=\angle A,2=B\angle 2=\angle B,请回答:

(1)A+B+ACB(1)\angle A+\angle B+\angle ACB的值为______^{\circ}
(2)(2)A=56\angle A=56^{\circ},B=40\angle B=40^{\circ},则ACF=______\angle ACF=\_\_\_\_\_\_^{\circ}
A=x\angle A=x^{\circ},B=y\angle B=y^{\circ},则ACF=\angle ACF=______;
我们把A\angle AB\angle BACB\angle ACB称为ABC\triangle ABC的内角;把ACF\angle ACF称为ABC\triangle ABC的外角,DEF\angle DEFDCE\triangle DCE的外角,每个三角形都有六个外角.
(3)(3)运用(1)(2)\left(1\right)\left(2\right)结论,解决问题:如图22,已知ABC\triangle ABC中,A=56\angle A=56^{\circ},BPBPCPCP分别平分ABC\angle ABCBCA\angle BCA,CQCQ平分外角ACF\angle ACFBPBP与点QQ,求BPC\angle BPC,BQC\angle BQC.
知识点:角的运算、三角形内角和定理、全等三角形的性质、全等三角形的判定、等腰三角形的性质、勾股定理、旋转的性质章节:第13章 三角形 / 13.1 三角形的概念

答案与解析

答案

(1)ACB+1+2=180\left(1\right)\because \angle ACB+\angle 1+\angle 2=180^{\circ}
1=A\because \angle 1=\angle A2=B\angle 2=\angle B
A+B+ACB=180\therefore \angle A+\angle B+\angle ACB=180^{\circ}
故答案为:180180
(2)A=56(2)\because \angle A=56^{\circ}B=40\angle B=40^{\circ}
1=56\therefore \angle 1=56^{\circ}2=40\angle 2=40^{\circ}
ACF=1+2=96\therefore \angle ACF=\angle 1+\angle 2=96^{\circ}
A=x\angle A=x^{\circ}B=y\angle B=y^{\circ}ACF=(x+y)\angle ACF=\left(x+y\right)^{\circ}
故答案为:9696(x+y)\left(x+y\right)^{\circ}
(3)A=56(3)\because \angle A=56^{\circ}A+ACB+ABC=180\angle A+\angle ACB+\angle ABC=180^{\circ}
ABC+ACB=18056=124\therefore \angle ABC+\angle ACB=180^{\circ}-56^{\circ}=124^{\circ}
BP\because BPCPCP分别平分ABC\angle ABCBCA\angle BCA
PBC=12ABC\therefore \angle PBC=\frac{1}{2}\angle ABCPCB=12ACB\angle PCB=\frac{1}{2}\angle ACB
PBC+PCB+BPC=180\because \angle PBC+\angle PCB+\angle BPC=180^{\circ}
BPC=18062=118\therefore \angle BPC=180^{\circ}-62^{\circ}=118^{\circ}
BP\because BP平分ABC\angle ABC
QBC=12ABC\therefore \angle QBC=\frac{1}{2}\angle ABC
CQ\because CQ平分外角ACF\angle ACF
QCF=12ACF\therefore \angle QCF=\frac{1}{2}\angle ACF
ACF=BAC+ABC\because \angle ACF=\angle BAC+\angle ABC
QCF=12(ABC+BAC)\therefore \angle QCF=\frac{1}{2}(\angle ABC+\angle BAC)
BQC=QCFQBC=12BAC=28\therefore \angle BQC=\angle QCF-\angle QBC=\frac{1}{2}\angle BAC=28^{\circ}.

解析

(1)ACB+1+2=180\left(1\right)\because \angle ACB+\angle 1+\angle 2=180^{\circ}
1=A\because \angle 1=\angle A2=B\angle 2=\angle B
A+B+ACB=180\therefore \angle A+\angle B+\angle ACB=180^{\circ}
故答案为:180180
(2)A=56(2)\because \angle A=56^{\circ}B=40\angle B=40^{\circ}
1=56\therefore \angle 1=56^{\circ}2=40\angle 2=40^{\circ}
ACF=1+2=96\therefore \angle ACF=\angle 1+\angle 2=96^{\circ}
A=x\angle A=x^{\circ}B=y\angle B=y^{\circ}ACF=(x+y)\angle ACF=\left(x+y\right)^{\circ}
故答案为:9696(x+y)\left(x+y\right)^{\circ}
(3)A=56(3)\because \angle A=56^{\circ}A+ACB+ABC=180\angle A+\angle ACB+\angle ABC=180^{\circ}
ABC+ACB=18056=124\therefore \angle ABC+\angle ACB=180^{\circ}-56^{\circ}=124^{\circ}
BP\because BPCPCP分别平分ABC\angle ABCBCA\angle BCA
PBC=12ABC\therefore \angle PBC=\frac{1}{2}\angle ABCPCB=12ACB\angle PCB=\frac{1}{2}\angle ACB
PBC+PCB+BPC=180\because \angle PBC+\angle PCB+\angle BPC=180^{\circ}
BPC=18062=118\therefore \angle BPC=180^{\circ}-62^{\circ}=118^{\circ}
BP\because BP平分ABC\angle ABC
QBC=12ABC\therefore \angle QBC=\frac{1}{2}\angle ABC
CQ\because CQ平分外角ACF\angle ACF
QCF=12ACF\therefore \angle QCF=\frac{1}{2}\angle ACF
ACF=BAC+ABC\because \angle ACF=\angle BAC+\angle ABC
QCF=12(ABC+BAC)\therefore \angle QCF=\frac{1}{2}(\angle ABC+\angle BAC)
BQC=QCFQBC=12BAC=28\therefore \angle BQC=\angle QCF-\angle QBC=\frac{1}{2}\angle BAC=28^{\circ}.

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