import{E as Y,c as I,ag as Z,ai as D,r as M,al as Q,q as X,k5 as x,jG as k,aM as A,C as P,ba as l,m as u,W as L,w as G,s as S,a as j,o as N,aa as _,l as ee}from"./index-hc8lvKav.js";const ie="Determine the value of an angle using the sum of the angles in a triangle",ae=!0,$e="mathLive",se=!0,re="AMCHybride",ne="23/08/2023",he="dc8c9",le="5G31";function de(){Y.call(this),this.sup="1-2-3-4-5",this.sup2=!1,this.sup3=!0,I.isHtml?this.spacingCorr=2:this.spacingCorr=1.5,I.isHtml?this.spacing=2:this.spacing=2,this.nbQuestions=5,this.correctionDetailleeDisponible=!0,this.nbCols=1,this.nbColsCorr=1;const E=function(W,C){return W+C<=180?180-(W+C):-1};this.nouvelleVersion=function(){this.listeQuestions=[],this.listeCorrections=[],this.autoCorrection=[];const W=Z({saisie:this.sup,min:1,max:12,melange:13,defaut:13,nbQuestions:this.nbQuestions,shuffle:!this.sup3});let C,R,F,e,i,t,o,c,T=0;for(let v=0,f,$,y,B=0;v<this.nbQuestions&&B<50;){const b=[],p=[];let H=[];const m=[];let w=D([0,1,2]);C=M(1,26),R=M(1,26,[C]),e=Q(C),i=Q(R),F=M(1,24,[C,R]),t=Q(F);const s=X(M(0,2),0,e),r=X(M(1,5),M(8,10),i);let a,n,g,h,d;switch(y="",$="",W[v]){case 1:w=[0],m.push(i+t+e),o=M(10,40),c=M(20,100),f=`$${e+i+t}$ is any triangle. Angle $\\widehat{${e+i+t}}$ measures $${o}\\degree$ and angle $\\widehat{${i+e+t}}$ measures $${c}\\degree$.<br>What is the measure of angle $ \\widehat{${i+t+e}}$ ?`,a=x(s,r,c,o),n=a.listePoints[2],n.nom=t,b.push(a,k(a)),p.push(a,k(a)),g=l(r,s,n,1.5,"","green",2,1,"none",.2,!0,!1,"",1.2),h=l(s,r,n,1.5,"","green",2,1,"none",.2,!0,!1,"",1.2),b.push(g,h),d=l(s,n,r,1.5,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),p.push(g,h,d),this.correctionDetaillee&&($+="In a triangle, the sum of the angles is equal to $180\\degree$.<br>",$+=`Hence: $\\widehat{${e+i+t}} + \\widehat{${i+t+e}} + \\widehat{${i+e+t}}=180\\degree$<br>`,$+=`Hence: $\\widehat{${i+t+e}}=180- \\left(\\widehat{${e+i+t}} + \\widehat{${i+e+t}}\\right)$.<br>Hence:`),$+=`$\\widehat{${i+t+e}}$= $180\\degree-\\left(${o}\\degree+${c}\\degree\\right)=180\\degree-${o+c}\\degree=${E(o,c)}\\degree$.<br>`,$+=`The angle $${u("\\widehat{"+i+t+e+"}","black")}$ measures $${u(E(o,c))}\\degree$.`,H=[E(o,c)];break;case 2:w=[0],m.push(i+t+e),o=90,c=M(20,70),f=`$${e+i+t}$ is a right triangle in $${i}$ and angle $\\widehat{${i+e+t}}$ measures $${c}\\degree$.<br>What is the measure of angle $\\widehat{${i+t+e}}$ ?`,a=x(s,r,c,o),n=a.listePoints[2],n.nom=t,b.push(a,k(a)),p.push(a,k(a)),g=l(r,s,n,1.5,"","green",2,1,"none",.2,!0,!1,"",1.2),h=l(s,r,n,1.5,"|","green",2),b.push(g,h),g=l(r,s,n,1.5,"","green",2,1,"none",.2,!0,!1,"",1.2),d=l(s,n,r,1.5,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),p.push(g,h,d),this.correctionDetaillee&&($+=`The triangle $${e+i+t}$ being right-angled in $${i}$, the angles $\\widehat{${i+e+t}}$ and $\\widehat{${i+t+e}}$ are complementary (their sum is equal to $90\\degree$).<br>`,$+=`Hence: $\\widehat{${i+t+e}}+ \\widehat{${i+e+t}}=90\\degree$<br>Hence:`),$+=`$\\widehat{${i+t+e}}=90\\degree-${c}\\degree=${90-c}\\degree$<br>`,$+=`The angle $${u("\\widehat{"+i+t+e+"}","black")}$ measures $${u(90-c)}\\degree$.`,H=[90-c];break;case 12:w=[0],m.push(i+t+e),o=M(30,150),c=L((180-o)/2,1),f=`$${e+i+t}$ is an isosceles triangle in $${i}$. The angle $\\widehat{${e+i+t}}$ measures $${o}\\degree$.<br>What is the measure of the angle $\\widehat{${i+t+e}}$?`,a=x(s,r,c,o),n=a.listePoints[2],n.nom=t,b.push(a,k(a),A("||","blue",P(a.listePoints[1],a.listePoints[2]),P(a.listePoints[1],a.listePoints[0]),2)),p.push(a,k(a),A("||","blue",P(a.listePoints[1],a.listePoints[2]),P(a.listePoints[1],a.listePoints[0]),2)),h=l(s,r,n,1.5,"","green",2,1,"none",.2,!0,!1,"",1.2),h.echelleMark=2,b.push(h),g=l(r,s,c,1.5,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),g.angleArrondi=1,h=l(s,r,-o,1.5,"","green",2,1,"none",.2,!0,!1,"",1.2),d=l(s,n,r,1.5,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),d.angleArrondi=1,p.push(g,h,d),this.correctionDetaillee&&($+="The angles at the base of an isosceles triangle are of equal measure.<br>",$+=`Hence: $\\widehat{${i+e+t}}=\\widehat{${i+t+e}}$.<br>`,$+="However, in a triangle, the sum of the angles is equal to $180\\degree$.<br>",$+=`Hence: $\\widehat{${e+i+t}}+ \\widehat{${i+t+e}}+ \\widehat{${i+e+t}}=180\\degree$.<br>`,$+=`Hence: $\\widehat{${e+i+t}}+2\\times \\widehat{${i+t+e}}=180\\degree$.<br>`,$+=`Let $${o}\\degree+2\\times \\widehat{${i+t+e}}=180\\degree$.<br>`,$+=`Hence $2\\times \\widehat{${i+t+e}}=180\\degree-${o}\\degree$.<br>Hence`),$+=`$\\widehat{${i+t+e}}=\\left(180\\degree-${o}\\degree\\right)\\div 2=${180-o}\\degree\\div 2=${G((180-o)/2)}\\degree$<br>`,$+=`The angle $${u("\\widehat{"+i+t+e+"}","black")}$ measures $${u(G((180-o)/2))}\\degree$.`,H=[L((180-o)/2,1)];break;case 3:w=[0],m.push(i+t+e),c=M(30,60,[90]),o=c,f=`$${e+i+t}$ is an isosceles triangle in $${t}$. The angle $\\widehat{${e+i+t}}$ measures $${o}\\degree$.<br>What is the measure of the angle $\\widehat{${i+t+e}}$?`,a=x(s,r,c,o),n=a.listePoints[2],n.nom=t,b.push(a,k(a),A("||","blue",P(a.listePoints[1],a.listePoints[2]),P(a.listePoints[2],a.listePoints[0]),2)),p.push(a,k(a),A("||","blue",P(a.listePoints[1],a.listePoints[2]),P(a.listePoints[2],a.listePoints[0]),2)),h=l(s,r,n,1.5,"","green",2,1,"none",.2,!0,!1,"",1.2),h.echelleMark=2,b.push(h),g=l(r,s,c,1.5,"","green",2,1,"none",.2,!0,!1,"",1.2),h=l(s,r,-o,1.5,"","green",2,1,"none",.2,!0,!1,"",1.2),d=l(s,n,r,1.5,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),p.push(g,h,d),this.correctionDetaillee&&($+="The two angles at the base of an isosceles triangle are equal.<br>",$+=`So $\\widehat{${e+i+t}}=\\widehat{${i+e+t}}=${c}\\degree$.<br>`,$+="Now, in a triangle, the sum of the angles is equal to $180\\degree$.<br>Hence:"),$+=`$\\widehat{${i+t+e}}=180\\degree-2\\times ${c}\\degree=180\\degree-${2*c}\\degree=${180-2*c}\\degree$.<br>`,$+=`The angle $${u("\\widehat{"+i+t+e+"}","black")}$ measures $${u(180-2*c)}\\degree$.`,H=[180-2*c];break;case 4:w=[0],m.push(e+t+i),f=`$${e+i+t}$ is a right triangle in $${e}$ and $\\widehat{${e+t+i}}=\\widehat{${e+i+t}}$.<br>What is the measure of the angle $\\widehat{${e+t+i}}$?`,o=45,c=90,a=x(s,r,c,o),n=a.listePoints[2],n.nom=t,b.push(a,k(a)),p.push(a,k(a)),g=l(r,s,n,1.5,"","green",2),g.echelleMark=2,h=l(s,r,n,1.5,"|","green",2),h.echelleMark=2,d=l(s,n,r,1.5,"|","green",2),d.echelleMark=2,b.push(g,h,d),h=l(s,r,-o,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),d=l(s,n,r,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),h.angleArrondi=0,d.angleArrondi=0,p.push(g,h,d),$+=`$\\widehat{${e+t+i}}=\\widehat{${e+i+t}}$.<br>`,this.correctionDetaillee?($+="However, in a triangle, the sum of the angles is equal to $180\\degree$.<br>",$+=`Hence: $2 \\times \\widehat{${e+t+i}} + 90\\degree=180\\degree$.<br>`,$+=`Hence: $2 \\times \\widehat{${e+t+i}}=180\\degree-90\\degree=90\\degree$.<br>`):$+=`Hence: $2 \\times \\widehat{${e+t+i}} + 90\\degree=180\\degree$.<br>`,$+=`Hence: $\\widehat{${e+t+i}}=90\\degree \\div 2=45\\degree$.<br>`,$+=`The angle $${u("\\widehat{"+e+t+i+"}","black")}$ measures $${u("45")}\\degree$.`,H=[45];break;case 6:w=D([0,1]),m.push(e+t+i,e+i+t),f=`$${e+i+t}$ is a right triangle in $${e}$. The angle $\\widehat{${e+i+t}}$ measures twice the angle $\\widehat{${e+t+i}}$.<br>`,f+=`What are the respective measures of the angles $\\widehat{${m[w[0]]}}$ and $\\widehat{${m[w[1]]}}$?`,o=60,c=90,a=x(s,r,c,o),n=a.listePoints[2],n.nom=t,b.push(a,k(a)),p.push(a,k(a)),g=l(r,s,n,1,"","green",2),g.echelleMark=2,h=l(s,r,n,1,"||","green",2),h.echelleMark=2,d=l(s,n,r,1,"|","green",1.5),d.echelleMark=2,b.push(g,h,d),h=l(s,r,-o,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),d=l(s,n,r,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),h.angleArrondi=0,d.angleArrondi=0,p.push(g,h,d),$+=`$\\widehat{${e+i+t}}=2\\times \\widehat{${e+t+i}}$.<br>`,this.correctionDetaillee&&($+=`Moreover, the triangle $${e+i+t}$ being right-angled in $${e}$, the angles $\\widehat{${e+t+i}}$ and $\\widehat{${e+i+t}}$ are complementary (their sum is equal to $90\\degree$).<br>`,$+=`Hence: $2 \\times \\widehat{${e+t+i}} + \\widehat{${e+t+i}}=90\\degree$.<br>`,$+=`Hence: $3 \\times \\widehat{${e+t+i}}=90\\degree$.<br>`),$+=`Hence: $\\widehat{${e+t+i}}=90\\degree \\div 3=30\\degree$.<br>`,$+=`$\\widehat{${e+i+t}}=2\\times \\widehat{${e+t+i}}=2\\times 30\\degree=60\\degree$<br>`,$+=`The angle $${u("\\widehat{"+e+t+i+"}","black")}$ measures $${u("30")}\\degree$ and the angle $${u("\\widehat{"+e+i+t+"}","black")}$ measures $${u("60")}\\degree$.`,H=[30,60];break;case 7:w=D([0,1]),m.push(e+t+i,e+i+t),f=`$${e+i+t}$ is a right triangle in $${e}$. The angle $\\widehat{${e+i+t}}$ measures a quarter of the angle $\\widehat{${e+t+i}}$.<br>`,f+=`What are the respective measures of the angles $\\widehat{${m[w[0]]}}$ and $\\widehat{${m[w[1]]}}$?`,o=18,c=90,a=x(s,r,c,o),n=a.listePoints[2],n.nom=t,b.push(a,k(a)),p.push(a,k(a)),g=l(r,s,n,1,"","green",2),g.echelleMark=2,h=l(s,r,n,1,"|","green",2),h.echelleMark=2,d=l(s,n,r,1,"||||","green",1.5),d.echelleMark=2,b.push(g,h,d),h=l(s,r,-o,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),d=l(s,n,r,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),h.angleArrondi=1,d.angleArrondi=1,p.push(g,h,d),$+=`Since $\\widehat{${e+i+t}}=\\dfrac{\\widehat{${e+t+i}}}{4}$, we deduce that $\\widehat{${e+t+i}}=4\\times \\widehat{${e+i+t}}$.<br>`,this.correctionDetaillee&&($+=`Moreover, the triangle $${e+i+t}$ being right-angled in $${e}$, the angles $\\widehat{${e+t+i}}$ and $\\widehat{${e+i+t}}$ are complementary (their sum is equal to $90\\degree$).<br>`,$+=`Hence: $4 \\times \\widehat{${e+i+t}} + \\widehat{${e+i+t}}=90\\degree$.<br>Hence`,$+=` $5 \\times \\widehat{${e+i+t}}=90\\degree$.<br>Hence`),$+=`$\\widehat{${e+i+t}}=90\\degree \\div 5=18\\degree$.<br>`,$+=`$\\widehat{${e+t+i}}=4\\times \\widehat{${e+i+t}}=4\\times 18\\degree=72\\degree$.<br>`,$+=`The angle $${u("\\widehat{"+e+t+i+"}","black")}$ measures $${u("72")}\\degree$ and the angle $${u("\\widehat{"+e+i+t+"}","black")}$ measures $${u("18")}\\degree$.`,H=[72,18];break;case 8:w=D([0,1]),m.push(e+t+i,e+i+t),f=`$${e+i+t}$ is a right triangle in $${e}$. The angle $\\widehat{${e+i+t}}$ is five times larger than the angle $\\widehat{${e+t+i}}$.<br>`,f+=`What are the respective measures of the angles $\\widehat{${m[w[0]]}}$ and $\\widehat{${m[w[1]]}}$?`,o=75,c=90,a=x(s,r,c,o),n=a.listePoints[2],n.nom=t,b.push(a,k(a)),p.push(a,k(a)),g=l(r,s,n,1.5,"","green",2),g.echelleMark=2,h=l(s,r,n,1.5,"|||||","green",2),h.echelleMark=2,d=l(s,n,r,1.5,"|","green",2),d.echelleMark=2,b.push(g,h,d),h=l(s,r,-o,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),d=l(s,n,r,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),h.angleArrondi=1,d.angleArrondi=1,p.push(g,h,d),$+=`$\\widehat{${e+i+t}}=5\\times \\widehat{${e+t+i}}$.<br>`,this.correctionDetaillee?($+=`Moreover, the triangle $${e+i+t}$ being right-angled in $${e}$, the angles $\\widehat{${e+t+i}}$ and $\\widehat{${e+i+t}}$ are complementary (their sum is equal to $90\\degree$).<br>`,$+=`Hence: $5 \\times \\widehat{${e+t+i}} + \\widehat{${e+t+i}}=90\\degree$.<br>`,$+=`Hence: $6 \\times \\widehat{${e+t+i}}=90\\degree$.<br>`):$+=`Hence: $5 \\times \\widehat{${e+t+i}} + \\widehat{${e+t+i}}=90\\degree$.<br>`,$+=`Hence: $\\widehat{${e+t+i}}=90\\degree \\div 6=15\\degree$<br>`,$+=`$\\widehat{${e+i+t}}=5\\times \\widehat{${e+t+i}}=5\\times 15\\degree=75\\degree$<br>`,$+=`The angle $${u("\\widehat{"+e+t+i+"}","black")}$ measures $${u("15")}\\degree$ and the angle $${u("\\widehat{"+e+i+t+"}","black")}$ measures $${u("75")}\\degree$.`,H=[15,75];break;case 5:w=[0],m.push(e+i+t),f=`$${e+i+t}$ is a triangle whose three angles are equal. What is the measure of each of these angles?`,a=x(s,r,60,60),n=a.listePoints[2],n.nom=t,g=l(r,s,60,1.5,"|","green",2),g.echelleMark=2,h=l(s,r,-60,1.5,"|","green",2),h.echelleMark=2,d=l(s,n,60,1.5,"|","green",2),d.echelleMark=2,b.push(a,g,h,d,k(a)),h=l(s,r,-60,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),d=l(s,n,60,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),p.push(a,g,h,d,k(a)),$+=`Since the triangle $${e+i+t}$ has three equal angles, $\\widehat{${e+i+t}}=\\widehat{${e+t+i}}=\\widehat{${i+e+t}}$<br>`,this.correctionDetaillee&&($+="However, in a triangle, the sum of the angles is equal to $180\\degree$.<br>",$+=`Hence $3\\times \\widehat{${e+i+t}}=180\\degree$.<br>`),$+=`Hence: $\\widehat{${e+i+t}}=180\\degree\\div 3=60\\degree$.<br>`,$+=`We therefore have $${u("\\widehat{"+e+i+t+"}","black")}=${u("\\widehat{"+e+t+i+"}","black")}=${u("\\widehat{"+i+e+t+"}","black")}=${u("60")}\\degree$.<br>`,$+=`The triangle $${e+i+t}$ is an equilateral triangle.`,H=[60];break;case 9:w=D([0,1]),m.push(e+t+i,e+i+t),f=`$${e+i+t}$ is a right triangle in $${e}$. The angle $\\widehat{${e+i+t}}$ measures one third of the angle $\\widehat{${e+t+i}}$.<br>`,f+=`What are the respective measures of the angles $\\widehat{${m[w[0]]}}$ and $\\widehat{${m[w[1]]}}$?`,o=22.5,c=90,a=x(s,r,c,o),n=a.listePoints[2],n.nom=t,b.push(a,k(a)),p.push(a,k(a)),g=l(r,s,n,1,"","green",2),g.echelleMark=2,h=l(s,r,n,1.5,"|","green",2),h.echelleMark=2,d=l(s,n,r,1.5,"|||","green",2),d.echelleMark=2,b.push(g,h,d),h=l(s,r,-o,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),d=l(s,n,r,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),h.angleArrondi=1,d.angleArrondi=1,p.push(g,h,d),$+=`Since $\\widehat{${e+i+t}}=\\dfrac{\\widehat{${e+t+i}}}{3}$, we deduce that $\\widehat{${e+t+i}}=3\\times \\widehat{${e+i+t}}$.<br>`,this.correctionDetaillee?($+=`Moreover, the triangle $${e+i+t}$ being right-angled in $${e}$, the angles $\\widehat{${e+t+i}}$ and $\\widehat{${e+i+t}}$ are complementary (their sum is equal to $90\\degree$).<br>`,$+=`Hence: $3 \\times \\widehat{${e+i+t}} + \\widehat{${e+i+t}}=90\\degree$.<br>Hence:`,$+=`$4 \\times \\widehat{${e+i+t}}=90\\degree$.<br>Hence:`):$+=`Hence: $3 \\times \\widehat{${e+i+t}} + \\widehat{${e+i+t}}=90\\degree$.<br>Hence:`,$+=`$\\widehat{${e+i+t}}=90\\degree \\div 4=22.5\\degree$.<br>`,$+=`$\\widehat{${e+t+i}}=3\\times \\widehat{${e+i+t}}=3\\times 22.5\\degree=67.5\\degree$<br>`,$+=`The angle $${u("\\widehat{"+e+t+i+"}","black")}$ measures $${u("67.5")}\\degree$ and the angle $${u("\\widehat{"+e+i+t+"}","black")}$ measures $${u("22.5")}\\degree$.`,H=[67.5,22.5];break;case 10:m.push(e+t+i,e+i+t,i+e+t),f=`$${e+i+t}$ is an isosceles triangle in $${e}$. Angle $\\widehat{${i+e+t}}$ measures two-thirds of angle $\\widehat{${e+i+t}}$.<br>`,f+=`What are the respective measures of the angles $\\widehat{${m[w[0]]}}$, $\\widehat{${m[w[1]]}}$ and $\\widehat{${m[w[2]]}}$?`,o=67.5,c=45,a=x(s,r,c,o),n=a.listePoints[2],n.nom=t,b.push(a,k(a),A("XX","blue",P(a.listePoints[0],a.listePoints[2]),P(a.listePoints[1],a.listePoints[0]),2)),p.push(a,k(a),A("XX","blue",P(a.listePoints[0],a.listePoints[2]),P(a.listePoints[1],a.listePoints[0]),2)),g=l(r,s,n,1.5,"||","green",2),g.echelleMark=2,h=l(s,r,n,1.5,"|||","green",2),h.echelleMark=2,b.push(g,h),g=l(r,s,c,1.5,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),h=l(s,r,-o,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),d=l(s,n,r,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),h.angleArrondi=1,d.angleArrondi=1,p.push(g,h,d),$+=`Since $\\widehat{${i+e+t}}=\\dfrac{2\\times \\widehat{${e+t+i}}}{3}$, we deduce that $\\widehat{${e+t+i}}=\\dfrac{3\\times \\widehat{${i+e+t}}}{2}$.<br>`,this.correctionDetaillee?($+=`Moreover, since the triangle $${e+i+t}$ is isosceles in $${e}$, $\\widehat{${e+t+i}}$ and $\\widehat{${e+i+t}}$ are equal, then $\\widehat{${e+i+t}}=\\widehat{${e+t+i} }=\\dfrac{3\\times \\widehat{${i+e+t}}}{2}$.<br>`,$+="However, in a triangle, the sum of the angles is equal to $180\\degree$.<br>",$+=`Hence: $\\widehat{${e+t+i}}+\\widehat{${e+i+t}}+\\widehat{${i+e+t}}=180\\degree$.<br>`,$+=`Hence: $\\dfrac{3 \\times \\widehat{${i+e+t}}}{2} + \\dfrac{3 \\times \\widehat{${i+e+t}}}{2} + \\widehat{${i+e+t}}= 180\\degree$.<br>`,$+=`Hence: $\\dfrac{3 \\times \\widehat{${i+e+t}}}{2}\\times 2 + \\widehat{${i+e+t}}=180\\degree$.<br>`,$+=`Hence: $3 \\times \\widehat{${i+e+t}} + \\widehat{${i+e+t}}=180\\degree$.<br>`,$+=`Hence: $4 \\times \\widehat{${i+e+t}}=180\\degree$.<br>Hence:`):$+=`Hence: $\\dfrac{3 \\times \\widehat{${i+e+t}}}{2} + \\dfrac{3 \\times \\widehat{${i+e+t}}}{2} + \\widehat{${i+e+t}}= 180\\degree$.<br>`,$+=`$\\widehat{${i+e+t}}=180\\degree \\div 4=45\\degree$.<br>`,$+=`$\\widehat{${e+t+i}}=\\widehat{${e+i+t}}=\\dfrac{3\\times \\widehat{${i+e+t}}}{2}=\\dfrac{3\\times 45\\degree}{2} =\\dfrac{135\\degree}{2}=67.5\\degree$<br>`,$+=`The angle $${u("\\widehat{"+e+t+i+"}","black")}$ measures $${u("67.5")}\\degree$, the angle $${u("\\widehat{"+e+i+t+"}","black")}$ measures $${u("67.5")}\\degree$ and the angle $${u("\\widehat{"+i+e+t+"}"," black")}$ measures $${u("45")}\\degree$.`,H=[67.5,67.5,45];break;case 11:m.push(e+t+i,e+i+t,i+e+t),f=`$${e+i+t}$ is an isosceles triangle in $${e}$. The angle $\\widehat{${e+i+t}}$ measures twice the angle $\\widehat{${i+e+t}}$.<br>`,f+=`What are the respective measures of the angles $\\widehat{${m[w[0]]}}$, $\\widehat{${m[w[1]]}}$ and $\\widehat{${m[w[2]]}}$?`,o=72,c=36,a=x(s,r,c,o),n=a.listePoints[2],n.nom=t,b.push(a,k(a),A("|||","blue",P(a.listePoints[0],a.listePoints[2]),P(a.listePoints[1],a.listePoints[0]),2)),p.push(a,k(a),A("|||","blue",P(a.listePoints[0],a.listePoints[2]),P(a.listePoints[1],a.listePoints[0]),2)),g=l(r,s,n,1.5,"|","green",2),g.echelleMark=2,h=l(s,r,n,1.5,"||","green",2),h.echelleMark=2,b.push(g,h),g=l(r,s,c,1.5,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),h=l(s,r,-o,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),d=l(s,n,r,.8,"","#f15929",2,1,"none",.2,!0,!1,"",1.2),p.push(g,h,d),$+=`We have $\\widehat{${e+i+t}}=2\\times \\widehat{${i+e+t}}$.<br>`,this.correctionDetaillee?($+=`Moreover, since the triangle $${e+i+t}$ is isosceles in $${e}$, $\\widehat{${e+t+i}}$ and $\\widehat{${e+i+t}}$ are equal, then $\\widehat{${e+t+i}}=\\widehat{${e+i+t} }=2\\times \\widehat{${i+e+t}}$.<br>`,$+="However, in a triangle, the sum of the angles is equal to $180\\degree$.<br>",$+=`Hence: $\\widehat{${e+t+i}}+\\widehat{${e+i+t}}+\\widehat{${i+e+t}}=180\\degree$.<br>`,$+=`Hence: $2 \\times \\widehat{${i+e+t}} + 2 \\times \\widehat{${i+e+t}} + \\widehat{${i+e+t}}=180\\degree$.<br>`,$+=`Hence: $4 \\times \\widehat{${i+e+t}} + \\widehat{${i+e+t}}=180\\degree$.<br>Hence`,$+=` $5 \\times \\widehat{${i+e+t}}=180\\degree$.<br>Hence`):$+=`Hence: $2 \\times \\widehat{${i+e+t}} + 2 \\times \\widehat{${i+e+t}} + \\widehat{${i+e+t}}=180\\degree$.<br>`,$+=`$\\widehat{${i+e+t}}=180\\degree \\div 5=36\\degree$.<br>`,$+=`$\\widehat{${e+i+t}}=\\widehat{${e+t+i}}=2\\times \\widehat{${i+e+t}}=2\\times 36\\degree=72\\degree$<br>`,$+=`The angle $${u("\\widehat{"+e+t+i+"}","black")}$ measures $${u("72")}\\degree$, the angle $${u("\\widehat{"+e+i+t+"}","black")}$ measures $${u("72")}\\degree$ and the angle $${u("\\widehat{"+i+e+t+"}"," black")}$ measures $${u("36")}\\degree$.`,H=[72,72,36];break}const q=[H[w[0]]];S(this,T,H[w[0]]),H.length>1&&(q.push(H[w[1]]),S(this,T+1,H[w[1]]),H.length>2&&(q.push(H[w[2]]),S(this,T+2,H[w[2]]))),this.interactif&&(f+="<br>"+j(this,T,"inline nospacebefore width15",{texte:`$\\widehat{${m[w[0]]}} = $`,texteApres:"$\\degree$"}),H.length>1&&(f+="<br>"+j(this,T+1,"inline nospacebefore width15",{texte:`$\\widehat{${m[w[1]]}} = $`,texteApres:"$\\degree$"}),H.length>2&&(f+="<br>"+j(this,T+2,"inline nospacebefore width15",{texte:`$\\widehat{${m[w[2]]}} = $`,texteApres:"$\\degree$"})))),T+=m.length;const J=k(a);b.push(J);const O=Math.min(s.x,r.x,n.x)-2,z=Math.max(s.x,r.x,n.x)+2,K=Math.min(s.y,r.y,n.y)-2,U=Math.max(s.y,r.y,n.y)+2,V={xmin:O,ymin:K,xmax:z,ymax:U,pixelsParCm:20,scale:1};this.sup2?f+="<br>"+N(V,b):y+=this.correctionDetaillee?"<br>"+N(V,b):"",y+=$,y+="<br>"+N(Object.assign(_(p)),p),I.isAmc&&(this.autoCorrection[v]={enonce:"",enonceAvant:!1,options:{barreseparation:!0},propositions:[{type:"AMCNum",propositions:[{texte:"",statut:"",reponse:{texte:f+`<br><br>Value of $\\widehat{${m[w[0]]}}$`,valeur:q[0],param:{digits:2,decimals:0,signe:!1,approx:0}}}]}]},H.length>1&&(this.autoCorrection[v].propositions[0].propositions[0].multicolsBegin=!0,this.autoCorrection[v].propositions.push({type:"AMCNum",propositions:[{texte:"",multicolsEnd:!0,statut:"",reponse:{texte:`Value of $\\widehat{${m[w[1]]}}$`,valeur:q[1],param:{digits:2,decimals:0,signe:!1,approx:0}}}]})),H.length>2&&(this.autoCorrection[v].propositions[1].propositions[0].multicolsEnd=!1,this.autoCorrection[v].propositions.push({type:"AMCNum",propositions:[{texte:"",multicolsEnd:!0,statut:"",reponse:{texte:`Value of $\\widehat{${m[w[2]]}}$`,valeur:q[2],param:{digits:2,decimals:0,signe:!1,approx:0}}}]}))),this.questionJamaisPosee(v,f)&&(this.listeQuestions.push(f),this.listeCorrections.push(y),v++),B++}ee(this)},this.besoinFormulaireTexte=["Different situations",["Numbers separated by hyphens","1: Any triangle with two known acute angles","2: Right triangle with a known acute angle","3: Isosceles triangle with a known base angle","4: Isosceles right triangle","5: Equilateral triangle","6: Right triangle with one acute angle twice the other (*)","7: Right triangle with one acute angle quarter to the other (*)","8: Right triangle with one acute angle five times the other (*)","9: Right triangle with one acute angle third of the other (*)","10: Right triangle with one acute angle two thirds from the other (*)","11: Isosceles triangle with one acute angle twice the other (*)","12: Isosceles triangle with known principal vertex angle (*)","13: Mixture","(*): More difficult question"].join(`
`)],this.besoinFormulaire2CaseACocher=["Add schema to questions"],this.besoinFormulaire3CaseACocher=["In order of different situations"]}export{se as amcReady,re as amcType,ne as dateDeModifImportante,de as default,ae as interactifReady,$e as interactifType,le as ref,ie as titre,he as uuid};
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