Ángulos 2

rand(2) rand(3) RAND_SWITCH2 + 10*RAND_SWITCH3

rand(15) + 40 rand(10) + 100 180 - Tri_Y - Tri_Z

Dado lo siguiente:

\color{green}{\angle{ABC}} = Tri_Y°
\color{purple}{\angle{BAC}} = Tri_Z°
\overline{DE} \parallel \overline{BC}

¿Cuánto vale \color{blue}{\angle{DAF}} {?}

¿Cuánto vale \color{blue}{\angle{CAE}} {?}

init({range:[[-5,5],[-3,5]],scale:[40,40]}),style({stroke:"#888",strokeWidth:2}),path([[-5,0],[5,0]]),path([[-5,-3],[5,5]]),path([[-5,3],[5,3]]),path([[-1.2,0],[-4,3]]),style({fill:"grey"},function(){label([-1.3,0],"A","below"),circle([-1.25,0],.05),label([-4,3],"B","above"),circle([-4,3],.05),label([2.5,3],"C","above"),circle([2.5,3],.05),label([-4,0],"D","above"),circle([-4,0],.05),label([3,0],"E","above"),circle([3,0],.05),label([-3.75,-2],"F","above"),circle([-3.75,-2],.05)}),arc([-4,3],.75,312,360,{stroke:"green"}),label([-3.2,3],"\\color{green}{"+Tri_Y+"°}","below right",{color:"green"}),arc([-1.3,0],.75,38,125,{stroke:"purple"}),label([-1.3,.7],"\\color{purple}{"+Tri_Z+"°}","above",{color:"purple"}),0==RAND_SWITCH2?(arc([-1,0],1,180,210,{stroke:"blue"}),LABEL=label([-3.3,0],"\\color{blue}{∠DAF} = {?}","below",{color:"blue"})):(arc([-1,0],1,0,45,{stroke:"blue"}),LABEL=label([1,0],"\\color{blue}{∠CAE} = {?}","above",{color:"blue"}))

NOTE: Los ángulos no están dibujados a escala.

Tri_X \Large{{}^\circ}

Recuerda que las medidas de los ángulos en un triángulo suman 180°. Resuelve para \color{orange}{\angle{BCA}} restando las medidas de los ángulos \color{purple}{\angle{BAC}} y \color{green}{\angle{ABC}} a 180°. Encontramos que \color{orange}{\angle{BCA}} = Tri_X°. arc([2.5,3],.75,180,220,{stroke:"orange"}),label([1.8,3],"\\color{orange}{"+Tri_X+"°}","below left",{color:"orange"})

Resuelve para \color{blue}{\angle{DAF}} usando el hecho de que es un ángulo correspondiente a \color{orange}{\angle{BCA}}. LABEL.remove(),LABEL=label([-3.3,0],"\\color{blue}{\\angle{DAF}}="+Tri_X+"°","below") Resuelve para \color{blue}{\angle{CAE}} usando el hecho de que es un ángulo alterno interno a \color{orange}{\angle{BCA}}. LABEL.remove(),LABEL=label([1,0],"\\color{blue}{∠CAE} = "+Tri_X+"°","above",{color:"blue"}) Eso significa que esos ángulos son iguales pues ambos están formados por el mismo conjunto de rectas paralelas \overline{BC} y \overline{DE}, y recta transversal \overline{CF}.

rand(20) + 100 180 - Y

Dado lo siguiente:

\overline{AB} \parallel \overline{CD} (La recta AB es paralela a la recta CD)
\color{purple}{\angle{EGB}} = X°.
\color{purple}{\angle{AGH}} = X°.
\color{purple}{\angle{BGH}} = 180 - X°

¿Cuánto vale \color{blue}{\angle{GHD}} {?}

¿Cuánto vale \color{blue}{\angle{CHF}} {?}

init({range:[[-6.5,6],[-5,5.2]],scale:[40,40]}),style({stroke:"#888",strokeWidth:2}),style({fill:"grey"},function(){path([[-5,2],[5,2]]),label([-5,2],"A","below"),circle([-5,2],.05),label([5,2],"B","below"),circle([5,2],.05),path([[-5,-2],[5,-2]]),label([-5,-2],"C","below"),circle([-5,-2],.05),label([5,-2],"D","below"),circle([5,-2],.05),path([[-5,-4],[4,5]]),label([4,5],"E","below"),circle([4,5],.05),label([-5,-4],"F","right"),circle([-5,-4],.05),label([1,2],"G","below right"),circle([1,2],.05),label([-3,-2],"H","below right"),circle([-3,-2],.05)}),0==RAND_SWITCH2?(arc([-2.9,-2],1,0,50,{stroke:"blue"}),LABEL=label([-2,-2],"\\color{blue}{\\angle{GHD}}= {?}","above right")):(arc([-2.9,-2],1,180,220,{stroke:"blue"}),LABEL=label([-4,-2.5],"\\color{blue}{\\angle{CHF}}= {?}","below left")),0==RAND_SWITCH3?(arc([1.2,2],1.5,0,50,{stroke:"purple"}),label([3.1,2],"\\color{purple}{\\angle{EGB}}="+X+"°","above right")):1==RAND_SWITCH3?(arc([1.2,2],1.5,180,220,{stroke:"purple"}),label([-1,2],"\\color{purple}{\\angle{AGH}}="+X+"°","below left")):(arc([1.2,2],1,220,0,{stroke:"purple"}),label([1.5,1.2],"\\color{purple}{\\angle{BGH}}="+(180-X)+"°","below right"))

NOTE: Los ángulos no están dibujados a escala.

X \Large{{}^\circ}

\color{blue}{\angle{GHD}} = \color{purple}{\angle{EGB}}. Sabemos esto pues hay 2 ángulos complementarios de un conjunto de rectas paralelas cortadas por una única recta.

Primero resuelve para \color{orange}{\angle{AGH}}. Sabemos que \color{orange}{\angle{AGH}} = \color{purple}{\angle{EGB}} pues los ángulos opuestos son iguales. arc([1,2],.88,180,225,{stroke:"orange"}),label([0,2],"\\color{orange}{"+X+"°}","below left")

\color{blue}{\angle{GHD}} = \color{purple}{\angle{AGH}} Sabemos esto pues son 2 ángulos alternos internos formados por un conjunto de rectas paralelas cortadas por una única recta.

\color{blue}{\angle{CHF}} = \color{purple}{\angle{AGH}}. Sabemos esto pues son dos ángulos correspondientes formados por un conjunto de rectas paralelas cortadas por otra única recta.

Primero resuelve para \color{orange}{\angle{AGH}}. sabemos que \color{orange}{\angle{AGH}} = 180° - \color{purple}{\angle{BGH}} , porque los ángulos a lo largo de una recta suman 180°. arc([1,2],.88,180,225,{stroke:"orange"}),label([0,2],"\\color{orange}{"+X+"°}","below left")

\color{blue}{\angle{GHD}} = \color{orange}{\angle{AGH}}. Sabemos que esos 2 ángulos son iguales pues son ángulos alternos internos de 2 rectas paralelas.

\color{blue}{\angle{CHF}} = \color{orange}{\angle{AGH}}. Sabemos que esos dos ángulos son iguales pues son ángulos correspondientes formados por rectas paralelas y una recta que las corta.

Por lo tanto, \angle{GHD} = X°. LABEL.remove(),label([-2,-2],"\\color{blue}{\\angle{GHD}}="+X+"°","above right") Por lo tanto, \angle{CHF} = X°. LABEL.remove(),label([-4,-2.5],"\\color{blue}{\\angle{CHF}}="+X+"°","below left")

rand(10) + 30 rand(10) + 100 180 - Tri1_Y - Tri1_Z

Dado lo siguiente:

\color{green}{\angle{BDC}°} = Tri1_Y
\color{orange}{\angle{DBE}°} = Tri1_X

¿Cuánto vale \color{blue}{\angle{RAND_SWITCH3 === 0 ? "CHE" : ( RAND_SWITCH3 === 1 ? "GHC" : "DHE" )}} {?}

init({range:[[-10,10],[-7,10]],scale:[25,25]}),style({stroke:"#888",strokeWidth:2}),path([[-8,5],[8,5],[-6,-6],[0,9],[0,9],[6,-6],[-8,5]]),label([-8,5],"A","left"),label([0,9],"B","above"),label([8,5],"C","right"),label([-6,-6],"D","below"),label([6,-6],"E","below"),label([-1.8,5],"F","above left"),label([1.8,5],"G","above right"),label([3.2,1.3],"H","below right"),label([0,-1.3],"I","below"),label([-3.2,1.3],"J","below left"),label([-5.5,-5.2],"\\color{green}{"+Tri1_Y+"°}","above right"),arc([-6,-6],1.3,40,71,{stroke:"green"}),label([0,7.4],"\\color{orange}{"+Tri1_X+"°}","below"),arc([0,9],1.3,245,290,{stroke:"orange"}),0==RAND_SWITCH3?(LABEL=label([4.8,1],"\\color{blue}{\\angle{CHE}}= {?}","right"),arc([3.2,1.3],1.7,287,35,{stroke:"blue"})):1==RAND_SWITCH3?(LABEL=label([4,2.5],"\\color{blue}{\\angle{GHC}}= {?}","above"),arc([3.2,1.3],1,35,118,{stroke:"blue"})):(LABEL=label([2.5,-.5],"\\color{blue}{\\angle{DHE}}= {?}","below"),arc([3.2,1.3],1.1,219,286,{stroke:"blue"}))

NOTE: Los ángulos no están dibujados a escala.

Tri1_Z 180-Tri1_Z \Large{{}^\circ}

\color{purple}{\angle{BHD}} = 180° - \color{green}{\angle{BDC}} - \color{orange}{\angle{DBE}} = Tri1_Z°. Esto es debido a que la suma de los ángulos internos de un triángulo suman 180°. arc([3.2,1.3],.75,118,220,{stroke:"purple"}),label([2.6,2],"\\color{purple}{"+Tri1_Z+"^\\circ}","below left")

\color{blue}{\angle{CHE}} = \color{purple}{\angle{BHD}}. Esto es debido a que son ángulos opuestos por el vértice y los ángulos opuestos son iguales.

\color{blue}{\angle{CHE}} = Tri1_Z°

LABEL.remove(),label([4.8,1],"\\color{blue}{\\angle{CHE}}="+Tri1_Z+"^\\circ","right")

\color{blue}{\angle{CHG}} \color{blue}{\angle{DHE}} = 180° - \color{purple}{\angle{BHD}}. Esto es porque los ángulos a lo largo de una recta suman 180°.

\color{blue}{\angle{GHC}} LABEL.remove(),label([4,2.5],"\\color{blue}{\\angle{GHC}}="+(180-Tri1_Z)+"^\\circ","above") \color{blue}{\angle{DHE}} LABEL.remove(),label([2.5,-.5],"\\color{blue}{\\angle{DHE}}="+(180-Tri1_Z)+"^\\circ","below") = 180 - Tri1_Z°

rand(10) + 30 rand(10) + 70 180 - Tri2_Y - Tri2_Z

Dado lo siguiente:

\color{green}{\angle{BDC}°} = Tri2_Y°
\color{orange}{\angle{AIC}°} = 180 - Tri2_Z°

\color{green}{\angle{GCH}°} = Tri2_Y°
\color{orange}{\angle{FGH}°} = 180 - Tri2_Z°

¿Cuánto vale \color{blue}{\angle{AJF}} {?}

¿Cuánto vale \color{blue}{\angle{IHE}} {?}

init({range:[[-10,10],[-7,10]],scale:[25,25]}),style({stroke:"#888",strokeWidth:2}),path([[-8,5],[8,5],[-6,-6],[0,9],[0,9],[6,-6],[-8,5]]),label([-8,5],"A","left"),label([0,9],"B","above"),label([8,5],"C","right"),label([-6,-6],"D","below"),label([6,-6],"E","below"),label([-1.8,5],"F","above left"),label([1.8,5],"G","above right"),label([3.2,1.3],"H","below right"),label([0,-1.3],"I","below"),label([-3.2,1.3],"J","below left"),0==RAND_SWITCH2?(label([-5.5,-5.2],"\\color{green}{"+Tri2_Y+"°}","above right"),arc([-6,-6],1.3,40,71,{stroke:"green"}),label([0,-.2],"\\color{orange}{"+(180-Tri2_Z)+"°}","above"),arc([0,-1],1,28,152,{stroke:"orange"}),LABEL=label([-3.7,2.5],"\\color{blue}{\\angle{AJF}}= {?}","above"),arc([-3.2,1.3],1,65,142,{stroke:"blue"})):(label([6.5,5],"\\color{green}{"+Tri2_Y+"°}","below left"),arc([8,5],1.3,180,220,{stroke:"green"}),label([1.3,4.5],"\\color{orange}{"+(180-Tri2_Z)+"°}","below left"),arc([1.6,5],.75,180,289,{stroke:"orange"}),LABEL=label([4,-.3],"\\color{blue}{\\angle{IHE}}= {?}","below left"),arc([3.1,1.2],.75,220,290,{stroke:"blue"}))

NOTE: Los ángulos no están dibujados a escala.

Tri2_X \Large{{}^\circ}

\color{purple}{\angle{DIJ}} = 180° - \color{orange}{\angle{AIC}}. Esto es debido a que los ángulos en ambos lados de una línea es de 180°. arc([0,-1.2],.75,143,220,{stroke:"purple"}),label([-.75,-1.2],"\\color{purple}{"+Tri2_Z+"°}","left")

\color{purple}{\angle{HGC}} = 180° - \color{orange}{\angle{FGH}}. Esto es porque los ángulos a lo largo de una recta o un plano son de 180°. arc([1.8,5],1,280,0,{stroke:"purple"}),label([2.5,4.3],"\\color{purple}{"+Tri2_Z+"°}","below right")

\color{teal}{\angle{DJI}} = 180° - \color{green}{\angle{BDC}} - \color{purple}{\angle{DIJ}}. Sabemos esto pues la suma de los ángulos dentro de un triángulo es 180°. arc([-3.2,1.3],.75,260,320,{stroke:"teal"}),label([-3.2,.5],"\\color{teal}{"+Tri2_X+"°}","below right")

\color{teal}{\angle{CHG}} = 180° - \color{green}{\angle{ACD}} - \color{purple}{\angle{HGC}}. Sabemos esto debido a que la suma de los ángulos dentro de un triángulo es de 180°. arc([3.2,1.3],.75,38,120,{stroke:"teal"}),label([3.4,1.78],"\\color{teal}{"+Tri2_X+"°}","above")

\color{blue}{\angle{AJF}} = \color{teal}{\angle{DJI}}. Sabemos que son iguales pues son ángulos opuestos por el vértice.

\color{blue}{\angle{IHE}} = \color{teal}{\angle{CHG}}. Sabemos que son iguales pues son ángulos opuestos.

Por lo tanto, \angle{AJF} = Tri2_X°. LABEL.remove(),label([-3.7,2.5],"\\color{blue}{\\angle{AJF}}="+Tri2_X+"°","above") Por lo tanto, \angle{IHE} = Tri2_X°. LABEL.remove(),label([4,-.3],"\\color{blue}{\\angle{IHE}}="+Tri2_X+"°","below left")