m∡A+m∡B+m∡C+m∡D=360°
NOTAM CU a=m∡A ; b=m∡B; c=m∡C ; d=m∡D
vom avea a+b+c+d=360
mai stim ca :
[tex]a= \frac{b+c+d}{3} [/tex]
[tex]b= \frac{c+d}{2} [/tex]
[tex] c=\frac{d}{2} [/tex]
inlocuim si vom avea
[tex]b= \frac{ \frac{d}{2} +d}{2} [/tex]
[tex]b= \frac{\frac{d+2d}{2} } {2} [/tex]
[tex]b= \frac{ \frac{3d}{2} }{2} [/tex]
[tex]b= \frac{3d}{2} * \frac{1}{2} = \frac{3d}{4} [/tex]
[tex]a= \frac{ \frac{3d}{4} + \frac{d}{2} +d}{3} [/tex]
[tex]a= \frac{\frac{3d+2d+4d}{4} }{3} [/tex]
[tex]a= \frac{ \frac{9d}{4} }{3} = \frac{9d}{4} * \frac{1}{3} = \frac{3d}{4} [/tex]
inlocuim in a+b+c+d=360 si vom avea
[tex] \frac{3d}{4} + \frac{3d}{4} + \frac{d}{2} +d=360[/tex]
[tex] \frac{3d+3d+2d+4d}{4} =360[/tex]
[tex] \frac{12d}{4} =360 [/tex]
3d=360
d=360:3
d=120 ⇒ m∡D=120°
[tex]c= \frac{d}{2} = \frac{120}{2} =60[/tex] ⇒m∡C=60°
[tex]b= \frac{3d}{4} = \frac{3*120}{4} =3*30=90[/tex] ⇒m∡B=90°
[tex]b= \frac{3d}{4} = \frac{3*120}{4} =3*30=90[/tex] ⇒m∡A=90°
