222:@0.031627:0.974518:0.062330:0.974518:0.062330:0.956333:0.031627:0.956333:0.000000:0.000000:0.000000
Geometría:@0.073089:0.043661:0.168070:0.043661:0.168070:0.024916:0.073089:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Ejemplo 3:@0.073089:0.136836:0.149461:0.136836:0.149461:0.120167:0.073089:0.120167:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Halla el valor de  , teniendo en cuenta que en la figura las rectas   y   son paralelas.:@0.073089:0.160620:0.680097:0.160620:0.680097:0.144908:0.073089:0.144908:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.191447:0.160620:0.198902:0.160620:0.198902:0.144214:0.191447:0.144214:0.000000
m n:@0.539400:0.160620:0.579362:0.160620:0.579362:0.144214:0.539400:0.144214:0.000000:0.000000:0.000000
Gráfica 4.6 rectas paralelas cortadas por una secante:@0.089269:0.190354:0.483939:0.190354:0.483939:0.173684:0.089269:0.173684:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 :@0.212516:0.230829:0.265087:0.230829:0.265087:0.214475:0.212516:0.214475:0.000000:0.000000:0.000000:0.000000
3x 12°:@0.240925:0.230290:0.286743:0.230290:0.286743:0.215734:0.240925:0.215734:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
:@0.158990:0.379316:0.186460:0.379316:0.186460:0.362962:0.158990:0.362962:0.000000:0.000000
45°:@0.186460:0.378777:0.208117:0.378777:0.208117:0.364221:0.186460:0.364221:0.000000:0.000000:0.000000
m:@0.448348:0.253768:0.462314:0.253768:0.462314:0.239213:0.448348:0.239213:0.000000
n:@0.453174:0.371756:0.462314:0.371756:0.462314:0.357201:0.453174:0.357201:0.000000
:@0.270576:0.254649:0.282983:0.254649:0.282983:0.236808:0.270576:0.236808:0.000000
Los ángulos   y   son correspondientes. Por lo tanto, :@0.506694:0.277554:0.924303:0.277554:0.924303:0.261842:0.506694:0.261842:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
α:@0.603479:0.278137:0.614726:0.278137:0.614726:0.260483:0.603479:0.260483:0.000000
ω:@0.634753:0.278137:0.647031:0.278137:0.647031:0.260483:0.634753:0.260483:0.000000
también son congruentes y tienen la misma medida. Es :@0.506676:0.294196:0.924279:0.294196:0.924279:0.278483:0.506676:0.278483:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
decir que :@0.506676:0.310838:0.579182:0.310838:0.579182:0.295125:0.506676:0.295125:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∡α:@0.579034:0.311420:0.605523:0.311420:0.605523:0.293766:0.579034:0.293766:0.000000:0.000000
=45º:@0.605430:0.310838:0.640349:0.310838:0.640349:0.295125:0.605430:0.295125:0.000000:0.000000:0.000000:0.000000
Los ángulos  y   son adyascentes y suplementarios. Por consiguiente, la suma de sus medidas es 180º.:@0.073074:0.425361:0.820166:0.425361:0.820166:0.409649:0.073074:0.409649:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
θ  α:@0.163325:0.425944:0.200158:0.425944:0.200158:0.408290:0.163325:0.408290:0.000000:0.000000:0.000000:0.000000
Es decir que:@0.073056:0.449145:0.160858:0.449145:0.160858:0.433433:0.073056:0.433433:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 ∡α+ ∡θ=:@0.160693:0.449728:0.251220:0.449728:0.251220:0.432074:0.160693:0.432074:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
180º.:@0.251128:0.449145:0.287187:0.449145:0.287187:0.433433:0.251128:0.433433:0.000000:0.000000:0.000000:0.000000:0.000000
Como :@0.073056:0.471154:0.121828:0.471154:0.121828:0.455442:0.073056:0.455442:0.000000:0.000000:0.000000:0.000000:0.000000
∡α=:@0.121725:0.471737:0.163454:0.471737:0.163454:0.454083:0.121725:0.454083:0.000000:0.000000:0.000000
45º, se tiene que :@0.163362:0.471154:0.287408:0.471154:0.287408:0.455442:0.163362:0.455442:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
45º:@0.441882:0.493163:0.465903:0.493163:0.465903:0.477450:0.441882:0.477450:0.000000:0.000000:0.000000
+ ∡θ=:@0.465811:0.493746:0.525341:0.493746:0.525341:0.476091:0.465811:0.476091:0.000000:0.000000:0.000000:0.000000:0.000000
180º:@0.525248:0.493163:0.558179:0.493163:0.558179:0.477450:0.525248:0.477450:0.000000:0.000000:0.000000:0.000000
Es decir, que :@0.073056:0.515172:0.167110:0.515172:0.167110:0.499459:0.073056:0.499459:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∡θ=:@0.166951:0.515754:0.208846:0.515754:0.208846:0.498100:0.166951:0.498100:0.000000:0.000000:0.000000
180º 45º:@0.209859:0.515172:0.283543:0.515172:0.283543:0.499459:0.209859:0.499459:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
–:@0.247096:0.515754:0.256300:0.515754:0.256300:0.498100:0.247096:0.498100:0.000000
∡θ=:@0.463161:0.537763:0.502847:0.537763:0.502847:0.520109:0.463161:0.520109:0.000000:0.000000:0.000000
135º:@0.503859:0.537181:0.536790:0.537181:0.536790:0.521468:0.503859:0.521468:0.000000:0.000000:0.000000:0.000000
Pero, :@0.073056:0.559190:0.112280:0.559190:0.112280:0.543477:0.073056:0.543477:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∡θ=:@0.112190:0.559772:0.154085:0.559772:0.154085:0.542118:0.112190:0.542118:0.000000:0.000000:0.000000
3:@0.155097:0.559190:0.164080:0.559190:0.164080:0.543477:0.155097:0.543477:0.000000
x:@0.165111:0.559190:0.172565:0.559190:0.172565:0.542784:0.165111:0.542784:0.000000
–:@0.173596:0.559772:0.182800:0.559772:0.182800:0.542118:0.173596:0.542118:0.000000
12º:@0.183812:0.559190:0.210043:0.559190:0.210043:0.543477:0.183812:0.543477:0.000000:0.000000:0.000000
Entonces, 3:@0.073056:0.581198:0.156003:0.581198:0.156003:0.565486:0.073056:0.565486:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.156975:0.581198:0.164429:0.581198:0.164429:0.564792:0.156975:0.564792:0.000000
–:@0.165460:0.581781:0.174664:0.581781:0.174664:0.564127:0.165460:0.564127:0.000000
12º 135º:@0.175676:0.581198:0.255509:0.581198:0.255509:0.565486:0.175676:0.565486:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
=:@0.202919:0.581781:0.218252:0.581781:0.218252:0.564127:0.202919:0.564127:0.000000
3:@0.466640:0.603207:0.475623:0.603207:0.475623:0.587495:0.466640:0.587495:0.000000
x:@0.475549:0.603207:0.483004:0.603207:0.483004:0.586801:0.475549:0.586801:0.000000
=:@0.484035:0.603790:0.499368:0.603790:0.499368:0.586135:0.484035:0.586135:0.000000
147º:@0.500380:0.603207:0.533311:0.603207:0.533311:0.587495:0.500380:0.587495:0.000000:0.000000:0.000000:0.000000
Luego :@0.073056:0.625216:0.123153:0.625216:0.123153:0.609503:0.073056:0.609503:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.123050:0.625216:0.130505:0.625216:0.130505:0.608810:0.123050:0.608810:0.000000
=:@0.131536:0.625799:0.146869:0.625799:0.146869:0.608144:0.131536:0.608144:0.000000
49º:@0.147881:0.625216:0.174112:0.625216:0.174112:0.609503:0.147881:0.609503:0.000000:0.000000:0.000000
Ejemplo 4:@0.073056:0.652578:0.149428:0.652578:0.149428:0.635908:0.073056:0.635908:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Halla la medida de los ángulos   y  , teniendo en cuenta la figura::@0.073056:0.679940:0.550939:0.679940:0.550939:0.664227:0.073056:0.664227:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
α β:@0.297422:0.680522:0.335746:0.680522:0.335746:0.662868:0.297422:0.662868:0.000000:0.000000:0.000000
Gráfica 4.7 rectas paralelas cortadas por una secante:@0.089273:0.709673:0.483943:0.709673:0.483943:0.693004:0.089273:0.693004:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
:@0.331838:0.885372:0.359449:0.885372:0.359449:0.867718:0.331838:0.867718:0.000000:0.000000
65°:@0.359449:0.884790:0.382827:0.884790:0.382827:0.869077:0.359449:0.869077:0.000000:0.000000:0.000000
2:@0.210976:0.884790:0.219959:0.884790:0.219959:0.869077:0.210976:0.869077:0.000000
:@0.221064:0.885372:0.243649:0.885372:0.243649:0.867718:0.221064:0.867718:0.000000:0.000000
14°:@0.244754:0.884790:0.270340:0.884790:0.270340:0.869077:0.244754:0.869077:0.000000:0.000000:0.000000
2:@0.243281:0.745095:0.252264:0.745095:0.252264:0.729383:0.243281:0.729383:0.000000
:@0.252264:0.745678:0.264542:0.745678:0.264542:0.728024:0.252264:0.728024:0.000000
Por ser ángulos correspondientes entre paralelas, se tiene :@0.506694:0.735151:0.924572:0.735151:0.924572:0.719438:0.506694:0.719438:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
que 2:@0.506694:0.751792:0.548092:0.751792:0.548092:0.736080:0.506694:0.736080:0.000000:0.000000:0.000000:0.000000:0.000000
β=β+:@0.547981:0.752375:0.600865:0.752375:0.600865:0.734721:0.547981:0.734721:0.000000:0.000000:0.000000:0.000000
65º:@0.600773:0.751792:0.624795:0.751792:0.624795:0.736080:0.600773:0.736080:0.000000:0.000000:0.000000
Luego, :@0.506694:0.772012:0.559582:0.772012:0.559582:0.756300:0.506694:0.756300:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
β=:@0.559468:0.772595:0.585956:0.772595:0.585956:0.754940:0.559468:0.754940:0.000000:0.000000
65º:@0.585864:0.772012:0.609885:0.772012:0.609885:0.756300:0.585864:0.756300:0.000000:0.000000:0.000000
2:@0.506694:0.792232:0.515677:0.792232:0.515677:0.776519:0.506694:0.776519:0.000000
α–:@0.517941:0.792814:0.540812:0.792814:0.540812:0.775160:0.517941:0.775160:0.000000:0.000000
14º y:@0.543242:0.792232:0.593659:0.792232:0.593659:0.776519:0.543242:0.776519:0.000000:0.000000:0.000000:0.000000:0.000000
  β+:@0.594791:0.792814:0.635429:0.792814:0.635429:0.775160:0.594791:0.775160:0.000000:0.000000:0.000000:0.000000
65º son suplementarios, luego, :@0.637859:0.792232:0.924263:0.792232:0.924263:0.776519:0.637859:0.776519:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
2:@0.506712:0.808874:0.515695:0.808874:0.515695:0.793161:0.506712:0.793161:0.000000
α–:@0.516726:0.809456:0.538189:0.809456:0.538189:0.791802:0.516726:0.791802:0.000000:0.000000
14º:@0.539201:0.808874:0.565432:0.808874:0.565432:0.793161:0.539201:0.793161:0.000000:0.000000:0.000000
+β+:@0.566444:0.809456:0.610382:0.809456:0.610382:0.791802:0.566444:0.791802:0.000000:0.000000:0.000000
65º 180º:@0.611394:0.808874:0.691227:0.808874:0.691227:0.793161:0.611394:0.793161:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
=:@0.638637:0.809456:0.653970:0.809456:0.653970:0.791802:0.638637:0.791802:0.000000
Pero, :@0.506731:0.829094:0.545955:0.829094:0.545955:0.813381:0.506731:0.813381:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
β=:@0.545865:0.829676:0.573457:0.829676:0.573457:0.812022:0.545865:0.812022:0.000000:0.000000
65º, reemplazando:@0.574469:0.829094:0.714004:0.829094:0.714004:0.813381:0.574469:0.813381:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
2:@0.506731:0.849313:0.515714:0.849313:0.515714:0.833601:0.506731:0.833601:0.000000
α–:@0.516744:0.849896:0.538207:0.849896:0.538207:0.832242:0.516744:0.832242:0.000000:0.000000
14º 65º 65º 180º:@0.539220:0.849313:0.706229:0.849313:0.706229:0.833601:0.539220:0.833601:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
+:@0.566462:0.849896:0.581796:0.849896:0.581796:0.832242:0.566462:0.832242:0.000000
+:@0.610051:0.849896:0.625384:0.849896:0.625384:0.832242:0.610051:0.832242:0.000000
=:@0.653639:0.849896:0.668972:0.849896:0.668972:0.832242:0.653639:0.832242:0.000000
2:@0.506749:0.869533:0.515732:0.869533:0.515732:0.853821:0.506749:0.853821:0.000000
α=:@0.516763:0.870116:0.544355:0.870116:0.544355:0.852462:0.516763:0.852462:0.000000:0.000000
180º 14º 130º:@0.545368:0.869533:0.672654:0.869533:0.672654:0.853821:0.545368:0.853821:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
+:@0.582605:0.870116:0.597939:0.870116:0.597939:0.852462:0.582605:0.852462:0.000000
–:@0.626194:0.870116:0.635397:0.870116:0.635397:0.852462:0.626194:0.852462:0.000000
2:@0.506749:0.889753:0.515732:0.889753:0.515732:0.874040:0.506749:0.874040:0.000000
α=:@0.516763:0.890336:0.544355:0.890336:0.544355:0.872681:0.516763:0.872681:0.000000:0.000000
64º:@0.545368:0.889753:0.571598:0.889753:0.571598:0.874040:0.545368:0.874040:0.000000:0.000000:0.000000
α=:@0.506749:0.910555:0.534342:0.910555:0.534342:0.892901:0.506749:0.892901:0.000000:0.000000
32º:@0.535354:0.909973:0.561584:0.909973:0.561584:0.894260:0.535354:0.894260:0.000000:0.000000:0.000000
MUESTRA EDITORIAL:@0.248902:0.947499:0.888171:0.113285:0.743922:0.050540:0.104653:0.884754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 Muestra editorial solo para fines didácticos – Prohibida su venta:@0.095585:0.659700:0.095585:0.341447:0.077558:0.341447:0.077558:0.659700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000