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【例】$∠A,∠B,∠C是\triangle ABC$的三个内角.
(1)已知$∠A= 50^{\circ },∠B= ∠C$,求$∠B,∠C$的度数；
(2)已知$∠A-∠B= 20^{\circ },∠C= 60^{\circ }$,求$∠A,∠B$的度数；
(3)已知$∠A= \frac {1}{2}∠B= \frac {1}{3}∠C$,求$∠A,∠B,∠C$的度数.
(1)已知$∠A= 50^{\circ },∠B= ∠C$,求$∠B,∠C$的度数；
$∠B=65^{\circ },∠C=65^{\circ }$
(2)已知$∠A-∠B= 20^{\circ },∠C= 60^{\circ }$,求$∠A,∠B$的度数；
$∠A=70^{\circ },∠B=50^{\circ }$
(3)已知$∠A= \frac {1}{2}∠B= \frac {1}{3}∠C$,求$∠A,∠B,∠C$的度数.
$∠A=30^{\circ },∠B=60^{\circ },∠C=90^{\circ }$
答案
【分析】利用三角形的内角和定理,即$∠A+∠B+∠C= 180^{\circ }$进行求解.
解：(1)$\because ∠A+∠B+∠C= 180^{\circ },∠A= 50^{\circ },$
$\therefore ∠B+∠C= 180^{\circ }-50^{\circ }=130^{\circ }$.
$\because ∠B= ∠C,$
$\therefore ∠B= ∠C= \frac {1}{2}×130^{\circ }=65^{\circ }$.
(2)$\because ∠A+∠B+∠C= 180^{\circ },∠C= 60^{\circ },$
$\therefore ∠A+∠B= 180^{\circ }-60^{\circ }=120^{\circ }$.
解方程组$\left\{\begin{array}{l} ∠A-∠B= 20^{\circ },\\ ∠A+∠B= 120^{\circ },\end{array}{\circ }, \right. 得\left\{\begin{array}{l} ∠A= 70^{\circ },\\ ∠B= 50^{\circ }.\end{array}{\circ }. \right. $
(3)$\because ∠A= \frac {1}{2}∠B= \frac {1}{3}∠C,$
$\therefore ∠B= 2∠A,∠C= 3∠A$.
$\because ∠A+∠B+∠C= 180^{\circ },$
$\therefore ∠A+2∠A+3∠A= 180^{\circ }$.
$\therefore ∠A= 30^{\circ }$.
$\therefore ∠B= 2∠A= 60^{\circ },∠C= 3∠A= 90^{\circ }$.
解：(1)$\because ∠A+∠B+∠C= 180^{\circ },∠A= 50^{\circ },$
$\therefore ∠B+∠C= 180^{\circ }-50^{\circ }=130^{\circ }$.
$\because ∠B= ∠C,$
$\therefore ∠B= ∠C= \frac {1}{2}×130^{\circ }=65^{\circ }$.
(2)$\because ∠A+∠B+∠C= 180^{\circ },∠C= 60^{\circ },$
$\therefore ∠A+∠B= 180^{\circ }-60^{\circ }=120^{\circ }$.
解方程组$\left\{\begin{array}{l} ∠A-∠B= 20^{\circ },\\ ∠A+∠B= 120^{\circ },\end{array}{\circ }, \right. 得\left\{\begin{array}{l} ∠A= 70^{\circ },\\ ∠B= 50^{\circ }.\end{array}{\circ }. \right. $
(3)$\because ∠A= \frac {1}{2}∠B= \frac {1}{3}∠C,$
$\therefore ∠B= 2∠A,∠C= 3∠A$.
$\because ∠A+∠B+∠C= 180^{\circ },$
$\therefore ∠A+2∠A+3∠A= 180^{\circ }$.
$\therefore ∠A= 30^{\circ }$.
$\therefore ∠B= 2∠A= 60^{\circ },∠C= 3∠A= 90^{\circ }$.
