import{E as Z,c as x,aj as O,q as F,r as m,o as P,l as _,aS as H,aF as G,aB as M,bo as z,cp as ee,bv as te,bb as d,I as R,cq as W,b5 as r,m as p,aH as J,W as ae,aL as K}from"./index-XCg2QAX4.js";import{c as se}from"./aleatoires-C4JUoVmT.js";import"./dateEtHoraires-1H8goc58.js";const re="Determining angles using equality cases",oe=!0,le="AMCHybride",ce="14/11/2020",he="10/12/2023",ue="d12db",de="5G30-1";function me(){Z.call(this),this.sup=1,this.nbQuestions=1,this.spacing=2,this.spacingCorr=x.isHtml?3:2,this.nouvelleVersion=function(){this.listeQuestions=[],this.listeCorrections=[],this.autoCorrection=[];const E=this.sup===3?O([1,2],this.nbQuestions):O([this.sup],this.nbQuestions);for(let y=0,L=0;y<this.nbQuestions&&L<50;L++){let $=[];const e=se(5,"Q",!0),t=F(0,0,e[0],"above left"),U=function(){const g=[],l=[];let o,h;x.isHtml?h="#f15929":h="black";let n=m(45,85);const C=m(8,10),v=m(7,10,C),c=H(G(F(1,0),t,m(-45,45)),t,n,C,e[2],"left"),a=m(45,70),s=H(t,c,a,v/C,e[4],"above right"),i=M(c,t),u=M(c,s),A=M(t,s,"","#f15929"),f=z(t,c,m(3,C-4),e[1],"above left"),b=ee(f,A,"","#f15929"),w=te(b,u,e[3],"above right"),D=d(s,t,c,1,"","black",2,1,x.isAmc?"none":"black",.1,!0),k=d(t,c,s,1,"","black",2,1,x.isAmc?"none":"black",.1,!0),T=R(t,f,c,w,s),Q=d(w,f,t,1,"","blue",2,1,"blue"),j=d(f,w,s,1,"","#f15929",2,1,"#f15929"),B=d(w,s,t,1,"","green",2,1,"green"),I=d(w,f,c,1,"","pink",2,1,"pink"),N=d(c,w,f,1,"","red",2,1,"red");x.isAmc?g.push(i,u,A,b,D,k,T):g.push(i,u,A,b,D,k,Q,j,B,I,N,T),n=Math.round(W(s,t,c)),l[0]=`In the figure below, the lines $(${e[0]}${e[4]})$ and $(${e[1]}${e[3]})$ are parallel.<br>`,l[0]+="The figure is not in full size.<br>",l[0]+=x.isAmc?"<br>":`We want to determine the measure of the angles of the quadrilateral $${e[0]}${e[1]}${e[3]}${e[4]}$ (all answers must be justified).`,l[1]=`${r(0)} Determine the measure of the angle $\\widehat{${e[3]}${e[1]}${e[2]}}$.`,l[2]=`${r(1)} Deduce the measure of the angle $\\widehat{${e[0]}${e[1]}${e[3]}}$.`,l[3]=`${r(2)} Determine the measure of the angle $\\widehat{${e[1]}${e[3]}${e[2]}}$.`,l[4]=`${r(3)} Deduce the measure of the angle $\\widehat{${e[1]}${e[3]}${e[4]}}$.`,l[5]=`${r(4)} Determine the measure of the angle $\\widehat{${e[3]}${e[4]}${e[0]}}$.`,x.isAmc||(l[6]=`${r(5)} Verify the following conjecture: “The sum of the angles of a quadrilateral is 360°.”`),o=`${r(0)} Since the lines $(${e[0]}${e[4]})$ and $(${e[1]}${e[3]})$ are parallel, the corresponding angles $\\widehat{${e[4]}${e[0]}${e[1]}}$ and $\\widehat{${e[3]}${e[1]}${e[2]}}$ are equal, therefore $\\widehat{${e[3]}${e[1]}${e[2]}}$ measure $${p(n)}\\degree$.<br>`,o+=`${r(1)} The angles $\\widehat{${e[0]}${e[1]}${e[3]}}$ and $\\widehat{${e[3]}${e[1]}${e[2]}}$ are supplementary adjacent, so $\\widehat{${e[0]}${e[1]}${e[3]}}$ measures $180\\degree-${n}\\degree=${p(180-n,h)}\\degree$. <br>`,o+=`${r(2)} In a triangle, the sum of the angles is $180\\degree$ so $\\widehat{${e[1]}${e[3]}${e[2]}}=180\\degree-\\widehat{${e[3]}${e[1]}${e[2]}}-\\widehat{${e[1]}${e[2]}${e[3]}}=180\\degree-${n}\\degree-${a}\\degree=${p(180-n-a)}\\degree$.<br>`,o+=`${r(3)} The angles $\\widehat{${e[1]}${e[3]}${e[2]}}$ and $\\widehat{${e[1]}${e[3]}${e[4]}}$ are supplementary adjacent, so $\\widehat{${e[1]}${e[3]}${e[4]}}$ measures $180\\degree-${180-n-a}\\degree=${p(n+a,h)}\\degree$. <br>`,o+=`${r(4)} Since the lines $(${e[0]}${e[4]})$ and $(${e[1]}${e[3]})$ are parallel, the corresponding angles $\\widehat{${e[1]}${e[3]}${e[2]}}$ and $\\widehat{${e[3]}${e[4]}${e[0]}}$ are equal, therefore $\\widehat{${e[3]}${e[4]}${e[0]}}$ measure $${p(180-n-a,h)}\\degree$.<br>`,o+=x.isAmc?"none":`${r(5)} The sum of the angles of the quadrilateral is therefore: $${n}\\degree+${p(180-n,"blue")}\\degree+${p(n+a,"blue")}\\degree+${p(180-n-a,"blue")}\\degree=${p(360)}\\degree$.<br>`,o+="$\\phantom{f/}$ The conjecture is well verified.";const V=[n,180-n,180-n-a,n+a,180-n-a],S={xmin:Math.min(t.x-8,c.x-8,s.x-8),ymin:Math.min(t.y-1,s.y-1,c.y-1),xmax:Math.max(s.x+2,t.x+2,c.x+2),ymax:Math.max(c.y+2,t.y+2,s.y+2),scale:.7};return[g,S,o,l,V]},X=function(){const g=[],l=[];let o,h,n,C,v,c,a,s,i,u,A,f,b,w,D;do a=G(F(m(8,10),m(1,3)),t,m(-40,40),e[1],"right"),A=J(t,a),f=m(6,8),b=m(40,60),s=H(a,t,b,f/A,e[2],"above left"),n=M(s,t),C=M(t,a),i=z(t,a,A/2+m(-1,1,0)/10,e[3],"below"),v=M(s,i),w=J(s,i),D=J(t,i),u=z(s,i,w*A/D,e[4],"below right"),c=M(a,u),h=ae(W(s,i,a),0);while(h===90);const k=d(i,t,s,1,"","black",2,1,"black",.2,!0),T=d(s,i,a,1,"","red",2,1,"red",.2,!0),Q=d(i,u,a,1,"","blue",2,1,"blue",.2,!0),j=d(t,s,i,1,"","green",2,1,"green",.2),B=d(a,i,u,1,"","#f15929",2,1,"#f15929",.2),I=d(u,a,i,1,"","pink",2,1,"pink",.2),N=d(t,i,s,1,"","gray",2,1,"gray",.2),V=R(t,a,s,i,u);g.push(n,C,v,c,k,T,Q,N,j,B,I,V),l[0]="The figure below is not in full size. All answers must be justified.",l[1]=`${r(0)} Determine the measure of the angle $\\widehat{${e[0]}${e[3]}${e[2]}}$.`,l[2]=`${r(1)} Deduce the measure of the angle $\\widehat{${e[3]}${e[2]}${e[0]}}$.`,l[3]=`${r(2)} Determine whether the lines $(${e[0]}${e[2]})$ and $(${e[4]}${e[1]})$ are parallel.`,l[4]=`${r(3)} If we consider that the segments $[${e[0]}${e[2]}]$ and $[${e[4]}${e[1]}]$ are of the same length, Determine the nature of the quadrilateral $${e[0]}${e[2]}${e[1]}${e[4]}$.`,o=`${r(0)} The angles $\\widehat{${e[0]}${e[3]}${e[2]}}$ and $\\widehat{${e[2]}${e[3]}${e[1]}}$ are supplementary adjacent, so $\\widehat{${e[0]}${e[3]}${e[2]}}$ measures $180\\degree-${h}\\degree=${p(180-h)}\\degree$. <br>`,o+=`${r(1)} In a triangle, the sum of the angles is $180\\degree$ so $\\widehat{${e[0]}${e[2]}${e[3]}}=180-\\widehat{${e[3]}${e[0]}${e[2]}}-\\widehat{${e[0]}${e[3]}${e[2]}}=180\\degree-${b}\\degree- ${180-h}\\degree=${p(-b+h)}\\degree$.<br>`,o+=`${r(2)} For the lines $(${e[0]}${e[2]})$ and $(${e[4]}${e[1]})$ cut by the secant $(${e[2]}${e[4]})$ the angles $\\widehat{${e[0]}${e[2]}${e[3]}}$ and $\\widehat{${e[1]}${e[4]}${e[3]}}$ are alternate-internal angles. <br>`,o+="$\\phantom{c/}$ Now, if alternate-internal angles are equal, then this means that the lines cut by the secant are parallel.<br>",o+=`$\\phantom{c/}$ The lines $(${e[0]}${e[2]})$ and $(${e[4]}${e[1]})$ are therefore ${K("parallel")}.<br>`,o+=`${r(3)} The lines $(${e[0]}${e[2]})$ and $(${e[4]}${e[1]})$ are parallel and the segments $[${e[0]}${e[2]}]$ and $[${e[4]}${e[1]}]$ are of the same length.<br>`,o+="$\\phantom{c/}$ Now, a quadrilateral which has parallel opposite sides of the same length is a parallelogram.<br>",o+=`$\\phantom{c/}$ So $${e[0]}${e[2]}${e[1]}${e[4]}$ is a ${K("parallelogram")} and $${e[3]}$ is its center.`;const S=[180-h,-b+h],Y={xmin:Math.min(t.x,a.x,s.x,i.x,u.x)-1,ymin:Math.min(t.y,a.y,s.y,i.y,u.y)-1,xmax:Math.max(t.x,a.x,s.x,i.x,u.x)+2,ymax:Math.max(t.y,a.y,s.y,i.y,u.y)+2};return[g,Y,o,l,S]};this.sup===3?E[y]=m(1,2):E[y]=this.sup,$=E[y]===1?U():X();let q="";for(let g=0;g<$[3].length;g++)q+=$[3][g]+"<br>",q+=g===0?P($[1],$[0]):"";x.isAmc&&(E[y]===1?this.autoCorrection[y]={enonce:$[3][0]+P($[1],$[0]),options:{barreseparation:!0,numerotationEnonce:!0},propositions:[{type:"AMCOpen",propositions:[{texte:"",numQuestionVisible:!1,statut:3,feedback:"",multicolsBegin:!0,enonce:$[3][1]+" Justify the answer."}]},{type:"AMCNum",propositions:[{texte:"",multicolsEnd:!0,reponse:{texte:`Value of angle $\\widehat{${e[3]}${e[1]}${e[2]}}$`,valeur:$[4][0],param:{signe:!1,digits:3,decimals:0}}}]},{type:"AMCNum",propositions:[{texte:"",reponse:{texte:$[3][2]+"<br>",valeur:$[4][1],param:{signe:!1,digits:3,decimals:0}}}]},{type:"AMCOpen",propositions:[{texte:"",numQuestionVisible:!1,statut:3,feedback:"",multicolsBegin:!0,enonce:$[3][3]+" Justify the answer."}]},{type:"AMCNum",propositions:[{texte:"",multicolsEnd:!0,reponse:{texte:`Value of angle $\\widehat{${e[1]}${e[3]}${e[2]}}$`,valeur:$[4][2],param:{signe:!1,digits:3,decimals:0}}}]},{type:"AMCNum",propositions:[{texte:"",reponse:{texte:$[3][4]+"<br>",valeur:$[4][3],param:{signe:!1,digits:3,decimals:0}}}]},{type:"AMCOpen",propositions:[{texte:"",numQuestionVisible:!1,statut:3,feedback:"",multicolsBegin:!0,enonce:$[3][5]+" Justify the answer."}]},{type:"AMCNum",propositions:[{texte:"",multicolsEnd:!0,reponse:{texte:`Value of angle $\\widehat{${e[3]}${e[4]}${e[0]}}$`,valeur:$[4][4],param:{signe:!1,digits:3,decimals:0}}}]}]}:this.autoCorrection[y]={enonce:$[3][0]+"<br>"+P($[1],$[0])+"<br>In this configuration, no AMC version developed.<br>",options:{barreseparation:!0,numerotationEnonce:!0},propositions:[]}),this.listeQuestions.push(q),this.listeCorrections.push($[2]),y++}_(this)},this.besoinFormulaireNumerique=["Figure type",3,`1: The trapeze
2: The butterfly
3: Random`]}export{oe as amcReady,le as amcType,he as dateDeModifImportante,ce as dateDePublication,me as default,de as ref,re as titre,ue as uuid};
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