Suppose $ABCD$ and $A_1B_1C_1D_1$ be quadrilaterals with corresponding angles equal. Also $AB=A_1B_1$, $AC=A_1C_1$, $BD=B_1D_1$. Are the quadrilaterals necessarily congruent?
Problem
Source: Sharygin finals 8.6
Tags: geometry
TheDarkPrince
01.08.2018 21:23
Answer is no. Construction: [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(5cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -10.72, xmax = 13.68, ymin = -6.06, ymax = 7.46; /* image dimensions */ pair X = (0.,6.), B = (-3.,0.), A = (3.,0.), C = (-1.228,3.544), D = (1.232,3.536), F = (1.8,2.4); draw(X--B--A--cycle, linewidth(1.)); /* draw figures */ draw(X--B, linewidth(1.)); draw(B--A, linewidth(1.)); draw(A--X, linewidth(1.)); draw(B--F, linewidth(1)); draw(A--(-1.8,2.4), linewidth(1.)); draw((-2.372,1.256)--(2.368,1.264), linewidth(1.)); draw(C--D, linewidth(1.)); /* dots and labels */ dot(X,dotstyle); dot(B,dotstyle); label("$B$", B, dir(180)); dot(A,dotstyle); label("$A$", A,dir(0)); dot(C,dotstyle); label("$C$", C,dir(180)); dot(D,dotstyle); label("$D$", D,dir(0)); dot((-1.8,2.4),linewidth(4.pt) + dotstyle); dot(F,linewidth(4.pt) + dotstyle); dot((-2.372,1.256),dotstyle); label("$C_1$", (-2.78,1.4), NE * labelscalefactor); dot((2.368,1.264),dotstyle); label("$D_1$", (2.44,1.46), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy]