Ángulos 1

rand(25) + 10 70 - rand( 15 ) 70 - rand( 15 ) 180 - Tri_Y - Tri_Z rand( 2 ) rand( 3 )

Dados los siguientes ángulos:

\overline{AB} \perp \overline{CD}, las rectas AB y CD sn perpendiculares.
\color{green}{\angle{CGE}} = ACCUTEANGLE° \color{green}{\angle{AGF}} = 90 - ACCUTEANGLE° \color{green}{\angle{DGF}} = ACCUTEANGLE°

¿Cuánto vale \color{blue}{\angle{AGF}} = {?} \color{blue}{\angle{CGE}} = {?} \color{blue}{\angle{BGE}} = {?}

init({range:[[-6,6],[-6,6]],scale:[40,40]}),style({stroke:"#888",strokeWidth:2}),path([[-5,0],[5,0]]),path([[0,-5],[0,5]]),path([[-2,-5],[2,5]]),style({fill:"grey"},function(){circle([5,0],.05),circle([-5,0],.05),circle([0,5],.05),circle([0,-5],.05),circle([2,5],.05),circle([-2,-5],.05)}),label([-5,0],"A","above"),label([5,0],"B","above"),label([0,5],"C","left"),label([0,-5],"D","right"),label([2,5],"E","right"),label([-2,-5],"F","left"),label([0,0],"G","below right"),0==RAND3?(label([.5,1.8],"\\color{green}{"+ACCUTEANGLE+"°}","above"),arc([0,0],1.2,70,90,{stroke:"green"}),ORIGINAL_LABEL=label([-1.2,-.75],"\\color{blue}{\\angle{AGF}}= {?}","below left"),arc([0,0],1.2,180,248,{stroke:"blue"})):1==RAND3?(label([-1.2,-.75],"\\color{green}{"+(90-ACCUTEANGLE)+"°}","below left"),arc([0,0],1.2,180,248,{stroke:"green"}),ORIGINAL_LABEL=label([.5,1.8],"\\color{blue}{\\angle{CGE}} = {?}","above"),arc([0,0],1.2,70,90,{stroke:"blue"})):(label([0,-2],"\\color{green}{"+ACCUTEANGLE+"°}","below left"),arc([0,0],1.2,248,270,{stroke:"green"}),ORIGINAL_LABEL=label([1.5,0],"\\color{blue}{\\angle{BGE}} = {?}","above right"),arc([0,0],1.2,0,70,{stroke:"blue"}))

NOTA: Los ángulos no necesariamente están dibujados a escala.

90-ACCUTEANGLE ACCUTEANGLE \Large{{}^\circ}

Como sabemos \overline{AB} \perp \overline{CD}, sabemos \color{purple}{\angle{CGB}} = 90° label([2.2,1.7],"\\color{purple}{90°}","above right"),arc([0,0],3,0,90,{stroke:"purple"})

\color{orange}{\angle{EGB}} = \color{green}{\angle{AGF}} = 90 - ACCUTEANGLE°, pues son ángulos opuestos por el vértice. Los ángulos opuestos por un vértice son congruentes (iguales). label([1.2,0],"\\color{orange}{"+(90-ACCUTEANGLE)+"°}","above right"),arc([0,0],1.2,0,68,{stroke:"orange"})

Como sabemos \overline{AB} \perp \overline{CD}, sabemos \color{purple}{\angle{AGD}} = 90° label([-2.2,-1.7],"\\color{purple}{90°}","below left"),arc([0,0],3,180,270,{stroke:"purple"})

\color{orange}{\angle{EGB}} = \color{purple}{90°} - \color{green}{\angle{CGE}} = 90 - ACCUTEANGLE° label([1.2,0],"\\color{orange}{"+(90-ACCUTEANGLE)+"°}","above right"),arc([0,0],1.2,0,68,{stroke:"orange"})

Como sabemos \overline{AB} \perp \overline{CD}, sabemos \color{purple}{\angle{CGB}} = 90° label([2.2,1.7],"\\color{purple}{90°}","above right"),arc([0,0],3,0,90,{stroke:"purple"})

\color{orange}{\angle{AGF}} = \color{purple}{90°} - \color{green}{\angle{DGF}} = 90 - ACCUTEANGLE° label([-1.2,0],"\\color{orange}{"+(90-ACCUTEANGLE)+"°}","below left"),arc([0,0],1.2,180,248,{stroke:"orange"})

\color{blue}{\angle{AGF}} = \color{orange}{\angle{EGB}} = 90 - ACCUTEANGLE°, pues son ángulos opuestos por el vértice. Los ángulos opuestos por un vértice son congruentes (iguales). ORIGINAL_LABEL.remove(),label([-1.2,-.75],"\\color{blue}{\\angle{AGF}}="+(90-ACCUTEANGLE)+"°","below left")

\color{blue}{\angle{CGE}} = \color{purple}{90°} - \color{orange}{\angle{EGB}} = ACCUTEANGLE° ORIGINAL_LABEL.remove(),label([.5,1.8],"\\color{blue}{\\angle{CGE}} = "+ACCUTEANGLE+"°","above")

\color{blue}{\angle{BGE}} = \color{orange}{\angle{AGF}} = 90 - ACCUTEANGLE°, pues son ángulos opuestos por el vértice. Los ángulos opuestos por un vértice son congruentes (iguales). ORIGINAL_LABEL.remove(),label([1.5,0],"\\color{blue}{\\angle{BGE}} = "+(90-ACCUTEANGLE)+"°","above right")

Dado lo siguiente:

\color{purple}{\angle{ABC}} = Tri_Z°
\color{green}{\angle{ACB}} = Tri_Y°

¿Cuánto mide \color{blue}{\angle{DAB}}?

\color{purple}{\angle{ABC}} = Tri_Z°
\color{green}{\angle{DAB}} = 180 - Tri_X°

¿Cuánto mide \color{blue}{\angle{ACB}}?

init({range:[[-9,6],[-3,5]],scale:[33,33]}),style({stroke:"#888",strokeWidth:2}),path([[-8,-2],[4,-2],[0,3],[-4,-2]]),style({fill:"grey"},function(){circle([-8,-2],.05),circle([4,-2],.05),circle([0,3],.05),circle([-4,-2],.05)}),label([-4,-2],"A","below"),label([0,3],"B","above"),label([4,-2],"C","below right"),label([-8,-2],"D","below"),0==RAND2?(label([3,-2],"\\color{green}{"+Tri_Y+"°}","above left"),arc([4,-2],1.2,130,180,{stroke:"green"}),label([0,1.5],"\\color{purple}{"+Tri_Z+"°}","below"),arc([0,3],1.5,230,310,{stroke:"purple"}),ORIGINAL_LABEL=label([-4.7,-2],"\\color{blue}{\\angle{DAB}}= {?}","above left"),arc([-4,-2],.75,50,180,{stroke:"blue"})):(label([-4.7,-2],"\\color{green}{"+(180-Tri_X)+"°}","above left"),arc([-4,-2],.75,50,180,{stroke:"green"}),label([0,1.5],"\\color{purple}{"+Tri_Z+"°}","below"),arc([0,3],1.5,230,310,{stroke:"purple"}),ORIGINAL_LABEL=label([2.8,-2],"\\color{blue}{\\angle{ACB}} = {?}","above left"),arc([4,-2],1.2,130,180,{stroke:"blue"}))

NOTA: Los ángulos no necesariamente están dibujados a escala.

Tri_Y + Tri_Z Tri_Y \Large{{}^\circ}

\color{orange}{\angle{BAC}} = 180° - \color{purple}{\angle{ABC}} - \color{green}{\angle{ACB}} = 180 - Tri_Y - Tri_Z° , esto es pues la suma de los ángulos internos de un triángulo es 180 grados. label([-3.3,-2],"\\color{orange}{"+Tri_X+"°}","above right"),arc([-4,-2],.75,0,49,{stroke:"orange"})

\color{orange}{\angle{BAC}} = 180° - \color{green}{\angle{DAB}} = 180 - Tri_Y - Tri_Z° , pues ángulos suplementarios en una recta suman 180 grados. label([-3.3,-2],"\\color{orange}{"+Tri_X+"°}","above right"),arc([-4,-2],.75,0,49,{stroke:"orange"})

\color{blue}{\angle{DAB}} = 180° - \color{orange}{\angle{BAC}} = Tri_Y + Tri_Z° , pues los ángulos suplementarios en una recta suman 180° ORIGINAL_LABEL.remove(),label([-4.7,-2],"\\color{blue}{\\angle{DAB}} = "+(Tri_Y+Tri_Z)+"°","above left")

\color{blue}{\angle{ACB}} = 180° - \color{orange}{\angle{BAC}} - \color{purple}{\angle{ABC}} = Tri_Y° , pues los ángulos internos de un triángulo suman 180°. ORIGINAL_LABEL.remove(),label([2.8,-2],"\\color{blue}{\\angle{ACB}} = "+Tri_Y+"°","above left")

Dado lo siguiente:

\overline{HI} \parallel \overline{JK}, Las restas HI y JK son paralelas.
\color{purple}{\angle{BAC}} = Tri_X°
\color{purple}{\angle{AKJ}} = Tri_Y°
\color{green}{\angle{AJK}} = Tri_Z°
\color{green}{\angle{AHI}} = Tri_Z°

¿Cuánto vale \color{blue}{\angle{AIH}} = {?} \color{blue}{\angle{AKJ}} = {?} \color{blue}{\angle{BAC}} = {?}

init({range:[[-9,9.5],[-5.7,8]],scale:[27,27]}),style({stroke:"#888",strokeWidth:2}),path([[-7.32,7.5],[8,-4]]),path([[-8,4],[4.03,-5]]),path([[-6,-4],[-4.24,7.5]]),path([[-6,-4],[9,-2]]),style({fill:"grey"},function(){circle([-7.32,7.5],.05),circle([8,-4],.05),circle([-8,4],.05),circle([4.03,-5],.05),label([-6,-4],"A","below"),circle([-6,-4],.05),label([-4.24,7.5],"B","right"),circle([-4.24,7.5],.05),label([9,-2],"C","above"),circle([9,-2],.05),label([-5.07,1.75],"H","below left"),label([-4.47,5.25],"J","above right"),label([1.25,-3],"I","below"),label([5.7,-2.3],"K","above")}),0==RAND3?(ORIGINAL_LABEL=label([0,-2.5],"\\color{blue}{\\angle{AIH}} = {?}","left"),arc([1.25,-3],.75,135,190,{stroke:"blue"}),label([-4.2,4.25],"\\color{green}{"+Tri_Z+"°}","below"),arc([-4.47,5.25],1,255,330,{stroke:"green"}),label([-5.5,-3.5],"\\color{purple}{"+Tri_X+"°}","above right"),arc([-6,-4],1,10,80,{stroke:"purple"})):1==RAND3?(ORIGINAL_LABEL=label([3.5,-2.6],"\\color{blue}{\\angle{AKJ}} = {?}","above"),arc([5.7,-2.3],.75,139,194,{stroke:"blue"}),label([-4.4,.65],"\\color{green}{"+Tri_Z+"°}","below"),arc([-5.07,1.75],1,257,326,{stroke:"green"}),label([-5.5,-3.5],"\\color{purple}{"+Tri_X+"°}","above right"),arc([-6,-4],1,10,80,{stroke:"purple"})):(ORIGINAL_LABEL=label([-5.5,-3.5],"\\color{blue}{\\angle{BAC}} = {?}","above right"),arc([-6,-4],1,10,80,{stroke:"blue"}),label([4.1,-2.6],"\\color{purple}{"+Tri_Y+"°}","above"),arc([5.7,-2.3],.75,139,194,{stroke:"purple"}),label([-4.4,.65],"\\color{green}{"+Tri_Z+"°}","below"),arc([-5.07,1.75],1,257,326,{stroke:"green"}))

NOTA: Los ángulos no necesariamente están dibujados a escala.

Tri_Y Tri_X \Large{{}^\circ}

\color{orange}{\angle{AHI}} = \color{green}{\angle{AJK}}, pues son ángulos correspondientes formados por dos recta paralelas y una transversal. Los ángulos correspondientes son congruentes (iguales). label([-4.6,.75],"\\color{orange}{"+Tri_Z+"°}","below"),arc([-5.07,1.75],1,260,325,{stroke:"orange"})

\color{orange}{\angle{AJK}} = \color{green}{\angle{AHI}}, pues son ángulos correspondientes formados por dos rectas paralelas y una transversal. Los ángulos correspondientes son congruentes (iguales). label([-4,4.25],"\\color{orange}{"+Tri_Z+"°}","below"),arc([-4.47,5.25],1,257,325,{stroke:"orange"})

\color{blue}{\angle{AIH}} = 180° - \color{orange}{\angle{AHI}} - \color{purple}{\angle{BAC}} = 180 - Tri_X - Tri_Z° , pues los 3 ángulos están contenidos en \triangle{AHI}. Los ángulos internos de un triángulo suman 180°. ORIGINAL_LABEL.remove(),label([0,-2.5],"\\color{blue}{\\angle{AIH}} = "+(180-Tri_X-Tri_Z)+"°","left")

ORIGINAL_LABEL.remove(),1===RAND3?label([3.3,-2.6],"\\color{blue}{\\angle{AKJ}} = "+Tri_Y+"°","above"):label([-5.5,-3.5],"\\color{blue}{\\angle{BAC}} = "+Tri_X+"°","above right") \color{blue}{\angle{AKJ}} = 180° - \color{orange}{\angle{AJK}} - \color{purple}{\angle{BAC}} = Tri_Y° \color{blue}{\angle{BAC}} = 180° - \color{orange}{\angle{AJK}} - \color{purple}{\angle{AKJ}} = Tri_X° , pues los 3 ángulos están contenidos en \triangle{AJK}. Los ángulos internos de un triángulo suman 180°.

Dado lo siguiente:

\overline{DE} \parallel \overline{FG}, Las rectas DE y FG son paralelas.
\overline{KL} \perp \overline{DE}, Las rectas KL y DE son perpendiculares.
\color{green}{\angle{GCJ}} = Tri_Y°
\color{green}{\angle{IAK}} = Tri_Y°

¿Cuánto vale \color{blue}{\angle{IAK}} = {?} \color{blue}{\angle{GCJ}} = {?}

init({range:[[-6,8],[-5,5]],scale:[35,35]}),style({stroke:"#888",strokeWidth:2}),path([[-5,2],[7,2]]),path([[-5,-2],[7,-2]]),path([[0,4],[0,-4]]),path([[-2,4],[6,-4]]),label([-5,2],"D","above left"),label([7,2],"E","above right"),label([-5,-2],"F","above left"),label([7,-2],"G","above right"),label([-2,4],"I","above left"),label([6,-4],"J","above right"),label([0,4],"K","above right"),label([0,-4],"L","below right"),label([0,2],"A","below left"),label([0,-2],"B","above left"),label([4,-2],"C","above right"),0==RAND2?(ORIGINAL_LABEL=label([0,3.5],"\\color{blue}{\\angle{IAK}} = {?}","above left"),arc([0,2],1,90,135,{stroke:"blue"}),label([4.75,-2],"\\color{green}{"+Tri_Y+"°}","below right"),arc([4,-2],.75,315,360,{stroke:"green"})):(label([0,3],"\\color{green}{"+Tri_Y+"°}","above left"),arc([0,2],1,90,135,{stroke:"green"}),ORIGINAL_LABEL=label([4.75,-2],"\\color{blue}{\\angle{GCJ} = {?}}","below right"),arc([4,-2],.75,315,360,{stroke:"blue"}))

NOTA: Los ángulos no necesariamente están dibujados a escala.

90 - Tri_Y \Large{{}^\circ}

\color{orange}{\angle{DAI}} = \color{green}{\angle{GCJ}} = Tri_Y°, como son ángulos alternos externos, formados por 2 rectas paralelas y una transversal, son congruentes (iguales). label([-.8,2],"\\color{orange}{"+Tri_Y+"°}","above left"),arc([0,2],1,135,180,{stroke:"orange"}) Alternativamente puedes asociar usando ángulos opuestos por el vértice y ángulos alternos internos para lograr el mismo resultado ( como se ve usando \color{pink}{pink}). label([1,2],"\\color{pink}{"+Tri_Y+"°}","below right"),arc([0,2],1,315,360,{stroke:"pink"}),label([3,-2],"\\color{pink}{"+Tri_Y+"°}","above left"),arc([4,-2],1,135,180,{stroke:"pink"})

\color{purple}{\angle{DAK}} = 90°, pues los ángulos formados por rectas perpendiculares son iguales a 90°. label([-1.68,2],"\\color{purple}{90°}","above left"),arc([0,2],1.65,90,180,{stroke:"purple"})

\color{blue}{\angle{IAK}} = 90° - \color{orange}{\angle{DAI}} = 90 - Tri_Y°, pues los ángulos \color{blue}{\angle{IAK}} y \color{orange}{\angle{DAI}} forman el ángulo \color{purple}{\angle{DAK}}. ORIGINAL_LABEL.remove(),label([0,3.5],"\\color{blue}{\\angle{IAK}} = "+(90-Tri_Y)+"°","above left")

\color{orange}{\angle{IAK}} = 90° - \color{green}{\angle{IAK}} = 90 - Tri_Y°, pues los ángulos \color{green}{\angle{IAK}} y \color{orange}{\angle{DAI}}, forman el ángulo \color{purple}{\angle{DAK}}. label([-.8,2],"\\color{orange}{"+(90-Tri_Y)+"°}","above left"),arc([0,2],1,135,180,{stroke:"orange"})

\color{blue}{\angle{GCJ}} = \color{orange}{\angle{DAI}} = 90 - Tri_Y°, como son ángulos alternos externos formados por 2 rectas paralelas y una recta transversal, son congruentes (iguales). Alternativamente, puedes emparejar usando ángulos opuestos por el vértice y ángulos alternos internos para obtener el mismo resultado (como se ve usando \color{pink}{pink}). label([1,2],"\\color{pink}{"+(90-Tri_Y)+"°}","below right"),arc([0,2],1,315,360,{stroke:"pink"}),label([3,-2],"\\color{pink}{"+(90-Tri_Y)+"°}","above left"),arc([4,-2],1,135,180,{stroke:"pink"}),ORIGINAL_LABEL.remove(),label([4.75,-2],"\\color{blue}{\\angle{GCJ} = "+(90-Tri_Y)+"°}","below right")