[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.61,0:00:04.35,Default,,0000,0000,0000,,I denne videoen skal vi se på noen
Dialogue: 0,0:00:04.36,0:00:07.44,Default,,0000,0000,0000,,rimelige enkle beviser for parallellogrammer.
Dialogue: 0,0:00:07.45,0:00:08.75,Default,,0000,0000,0000,,Vi starter med å bevise,
Dialogue: 0,0:00:08.76,0:00:10.87,Default,,0000,0000,0000,,at det motstående sider
Dialogue: 0,0:00:10.88,0:00:13.97,Default,,0000,0000,0000,,i det her parallellogrammet ABDC er like lange.
Dialogue: 0,0:00:13.98,0:00:19.57,Default,,0000,0000,0000,,AB er altså like lange som DC, og AD er lik med BC.
Dialogue: 0,0:00:19.58,0:00:21.76,Default,,0000,0000,0000,,La oss tegne
Dialogue: 0,0:00:21.77,0:00:24.16,Default,,0000,0000,0000,,en diagonal.
Dialogue: 0,0:00:24.59,0:00:27.58,Default,,0000,0000,0000,,Den her diagonalen krysser 2 sett parallelle linjer,
Dialogue: 0,0:00:27.59,0:00:31.01,Default,,0000,0000,0000,,så vi kan også se på den
Dialogue: 0,0:00:31.02,0:00:32.34,Default,,0000,0000,0000,,som en transversal.
Dialogue: 0,0:00:32.35,0:00:34.17,Default,,0000,0000,0000,,.
Dialogue: 0,0:00:34.18,0:00:35.39,Default,,0000,0000,0000,,.
Dialogue: 0,0:00:35.76,0:00:37.96,Default,,0000,0000,0000,,.
Dialogue: 0,0:00:38.45,0:00:40.84,Default,,0000,0000,0000,,.
Dialogue: 0,0:00:41.12,0:00:44.97,Default,,0000,0000,0000,,Vi kan altså se på diagonalen DB
Dialogue: 0,0:00:44.98,0:00:48.88,Default,,0000,0000,0000,,som en transversal til de parallelle linjene AB og DC.
Dialogue: 0,0:00:48.89,0:00:54.34,Default,,0000,0000,0000,,Vi kan kalle den her vinkelen for ABD
Dialogue: 0,0:00:54.35,0:00:55.60,Default,,0000,0000,0000,,og se litt nærmere på den.
Dialogue: 0,0:00:55.61,0:00:58.43,Default,,0000,0000,0000,,Den vil være kongruent, altså lik, med vinkel BCD,
Dialogue: 0,0:00:58.44,0:01:03.40,Default,,0000,0000,0000,,fordi de er tilsvarende innvendige vinkler.
Dialogue: 0,0:01:03.41,0:01:05.32,Default,,0000,0000,0000,,Vi vet altså,
Dialogue: 0,0:01:05.33,0:01:10.64,Default,,0000,0000,0000,,at vinkel ABD er kongruent
Dialogue: 0,0:01:10.65,0:01:13.62,Default,,0000,0000,0000,,med vinkel BCD.
Dialogue: 0,0:01:15.95,0:01:19.73,Default,,0000,0000,0000,,Vi kan se at diagonalen DB som en
Dialogue: 0,0:01:19.74,0:01:22.43,Default,,0000,0000,0000,,transversal til de her 2 parallelle linjene,
Dialogue: 0,0:01:22.44,0:01:27.36,Default,,0000,0000,0000,,altså de andre parallelle linjene AD og BC.
Dialogue: 0,0:01:27.37,0:01:31.26,Default,,0000,0000,0000,,Nå kan vi se,
Dialogue: 0,0:01:31.27,0:01:40.52,Default,,0000,0000,0000,,at vinkel DBC er kongruent med
Dialogue: 0,0:01:40.53,0:01:49.65,Default,,0000,0000,0000,,vinkel ABD av nøyaktig samme årsak.
Dialogue: 0,0:01:49.66,0:01:52.86,Default,,0000,0000,0000,,.
Dialogue: 0,0:01:53.19,0:01:54.26,Default,,0000,0000,0000,,.
Dialogue: 0,0:01:54.27,0:02:03.08,Default,,0000,0000,0000,,Tilsvarende innvendige vinkler en kongruente,
Dialogue: 0,0:02:03.09,0:02:06.41,Default,,0000,0000,0000,,når en transversal krysser 2 parallelle linjer.
Dialogue: 0,0:02:06.72,0:02:09.63,Default,,0000,0000,0000,,Trekanten ABD og trekant CDB
Dialogue: 0,0:02:09.64,0:02:16.12,Default,,0000,0000,0000,,deler den her siden.
Dialogue: 0,0:02:16.13,0:02:18.02,Default,,0000,0000,0000,,Den er selvfølgelig lik med seg selv.
Dialogue: 0,0:02:18.03,0:02:20.03,Default,,0000,0000,0000,,Hvordan kan vi bruke den kunnskapen?
Dialogue: 0,0:02:20.04,0:02:23.25,Default,,0000,0000,0000,,Begge de her trekantene
Dialogue: 0,0:02:23.26,0:02:26.78,Default,,0000,0000,0000,,har den lyserøde vinkelen og den her siden til felles,
Dialogue: 0,0:02:26.79,0:02:28.86,Default,,0000,0000,0000,,og de har også den grønne vinkelen.
Dialogue: 0,0:02:28.87,0:02:32.51,Default,,0000,0000,0000,,Lyserød vinkel, side og grønn vinkel.
Dialogue: 0,0:02:32.52,0:02:35.83,Default,,0000,0000,0000,,Vi har vist med vinkel-side-vinkel,
Dialogue: 0,0:02:35.84,0:02:37.91,Default,,0000,0000,0000,,at de 2 trekantene er kongruente.
Dialogue: 0,0:02:37.92,0:02:39.45,Default,,0000,0000,0000,,.
Dialogue: 0,0:02:39.46,0:02:44.16,Default,,0000,0000,0000,,Vi har vist i tidligere videoer,
Dialogue: 0,0:02:50.04,0:03:00.24,Default,,0000,0000,0000,,at vi kan gjøre det.
Dialogue: 0,0:03:00.45,0:03:03.16,Default,,0000,0000,0000,,.
Dialogue: 0,0:03:03.41,0:03:09.34,Default,,0000,0000,0000,,Det vet vi ut fra vinkel-side-vinkelkongruens.
Dialogue: 0,0:03:09.35,0:03:10.94,Default,,0000,0000,0000,,Hva forteller det oss?
Dialogue: 0,0:03:10.95,0:03:14.79,Default,,0000,0000,0000,,Hvis 2 trekanter er kongruente,
Dialogue: 0,0:03:14.80,0:03:17.96,Default,,0000,0000,0000,,vil alle egenskapene i de 2 trekantene være kongruente.
Dialogue: 0,0:03:17.97,0:03:24.28,Default,,0000,0000,0000,,Side DC er lik med side BA.
Dialogue: 0,0:03:24.29,0:03:27.94,Default,,0000,0000,0000,,Side DC i den nederste er det samme som
Dialogue: 0,0:03:27.95,0:03:28.95,Default,,0000,0000,0000,,side BA i den øverste.
Dialogue: 0,0:03:28.96,0:03:31.04,Default,,0000,0000,0000,,De er kongruente.
Dialogue: 0,0:03:31.05,0:03:32.42,Default,,0000,0000,0000,,.
Dialogue: 0,0:03:32.43,0:03:39.07,Default,,0000,0000,0000,,DC er lik med BA,
Dialogue: 0,0:03:39.08,0:03:46.99,Default,,0000,0000,0000,,fordi de er tilsvarende sider i kongruente trekanter.
Dialogue: 0,0:03:47.00,0:03:51.30,Default,,0000,0000,0000,,Vi kan bruke samme logikk til å si,
Dialogue: 0,0:03:51.31,0:03:54.92,Default,,0000,0000,0000,,at AD svarer til CB.
Dialogue: 0,0:03:58.44,0:04:02.73,Default,,0000,0000,0000,,Tilsvarende sider i kongruente
Dialogue: 0,0:04:02.74,0:04:05.14,Default,,0000,0000,0000,,trekanter er nemlig like.
Dialogue: 0,0:04:05.15,0:04:06.27,Default,,0000,0000,0000,,Nå er vi ferdige.
Dialogue: 0,0:04:06.59,0:04:09.67,Default,,0000,0000,0000,,Vi har bevist, at de motstående sidene er kongruente.
Dialogue: 0,0:04:09.68,0:04:11.34,Default,,0000,0000,0000,,La oss prøve det omvendt.
Dialogue: 0,0:04:13.24,0:04:16.41,Default,,0000,0000,0000,,Vi har en firkant, og vi vet,
Dialogue: 0,0:04:16.42,0:04:18.89,Default,,0000,0000,0000,,at de motstående sidene er kongruente.
Dialogue: 0,0:04:18.90,0:04:22.13,Default,,0000,0000,0000,,Kan vi bevise, at det er et parallellogram?
Dialogue: 0,0:04:22.14,0:04:24.53,Default,,0000,0000,0000,,Det er det samme beviset, nå baklengs.
Dialogue: 0,0:04:24.54,0:04:26.74,Default,,0000,0000,0000,,Vi tegner en diagonal her.
Dialogue: 0,0:04:26.75,0:04:28.87,Default,,0000,0000,0000,,Vi vet jo masse om trekanter.
Dialogue: 0,0:04:28.88,0:04:30.70,Default,,0000,0000,0000,,.
Dialogue: 0,0:04:31.63,0:04:33.12,Default,,0000,0000,0000,,.
Dialogue: 0,0:04:34.02,0:04:35.65,Default,,0000,0000,0000,,.
Dialogue: 0,0:04:35.66,0:04:37.83,Default,,0000,0000,0000,,.
Dialogue: 0,0:04:37.84,0:04:38.59,Default,,0000,0000,0000,,Vi vet selvfølgelig,
Dialogue: 0,0:04:38.60,0:04:42.42,Default,,0000,0000,0000,,at CB er lik med seg selv.
Dialogue: 0,0:04:42.43,0:04:44.08,Default,,0000,0000,0000,,.
Dialogue: 0,0:04:44.09,0:04:46.86,Default,,0000,0000,0000,,Det er jo den samme linjen.
Dialogue: 0,0:04:46.87,0:04:48.46,Default,,0000,0000,0000,,Vi har nå oppdelt firkanten i 2 trekanter.
Dialogue: 0,0:04:48.47,0:04:53.11,Default,,0000,0000,0000,,Vi har trekant ACB
Dialogue: 0,0:04:53.12,0:04:56.41,Default,,0000,0000,0000,,og trekant DBC.
Dialogue: 0,0:04:56.42,0:05:00.53,Default,,0000,0000,0000,,Alle 3 sidene i de 2 trekantene
Dialogue: 0,0:05:00.54,0:05:01.75,Default,,0000,0000,0000,,er lik med hverandre.
Dialogue: 0,0:05:01.76,0:05:04.89,Default,,0000,0000,0000,,Vi vet altså, at de er side-side-sidekongruente.
Dialogue: 0,0:05:04.90,0:05:11.79,Default,,0000,0000,0000,,.
Dialogue: 0,0:05:11.80,0:05:21.65,Default,,0000,0000,0000,,Trekanten ABC er kongruent med trekant DBC.
Dialogue: 0,0:05:24.00,0:05:30.55,Default,,0000,0000,0000,,.
Dialogue: 0,0:05:30.56,0:05:32.32,Default,,0000,0000,0000,,Hva forteller det oss?
Dialogue: 0,0:05:32.33,0:05:34.72,Default,,0000,0000,0000,,Det forteller oss, at alle de tilsvarende
Dialogue: 0,0:05:34.73,0:05:36.13,Default,,0000,0000,0000,,vinklene er kongruente.
Dialogue: 0,0:05:36.35,0:05:42.15,Default,,0000,0000,0000,,Vinkel ABC er altså
Dialogue: 0,0:05:49.25,0:05:52.86,Default,,0000,0000,0000,,kongruent med vinkel DCB.
Dialogue: 0,0:05:54.49,0:06:02.60,Default,,0000,0000,0000,,De tilsvarende vinklene er jo like,
Dialogue: 0,0:06:02.61,0:06:06.79,Default,,0000,0000,0000,,når 2 trekanter er kongruente.
Dialogue: 0,0:06:06.80,0:06:08.98,Default,,0000,0000,0000,,.
Dialogue: 0,0:06:08.99,0:06:12.28,Default,,0000,0000,0000,,ABC er kongruent med DCB.
Dialogue: 0,0:06:12.29,0:06:15.18,Default,,0000,0000,0000,,.
Dialogue: 0,0:06:15.19,0:06:18.23,Default,,0000,0000,0000,,Her har vi en lang linje,
Dialogue: 0,0:06:18.24,0:06:23.03,Default,,0000,0000,0000,,som krysser AB og CD, og vi kan se,
Dialogue: 0,0:06:23.04,0:06:26.77,Default,,0000,0000,0000,,at de er tilsvarende innvendige vinkler.
Dialogue: 0,0:06:26.78,0:06:27.76,Default,,0000,0000,0000,,De er kongruente.
Dialogue: 0,0:06:27.77,0:06:30.84,Default,,0000,0000,0000,,.
Dialogue: 0,0:06:30.85,0:06:33.95,Default,,0000,0000,0000,,Derfor må AB være parallell med CD.
Dialogue: 0,0:06:33.96,0:06:36.83,Default,,0000,0000,0000,,.
Dialogue: 0,0:06:36.84,0:06:47.49,Default,,0000,0000,0000,,Vi vet altså,
Dialogue: 0,0:06:47.50,0:06:51.53,Default,,0000,0000,0000,,at AB er parallell med CD.
Dialogue: 0,0:06:51.54,0:06:53.87,Default,,0000,0000,0000,,Nå kan vi bruke akkurat samme logikk til å si,
Dialogue: 0,0:06:57.02,0:07:04.59,Default,,0000,0000,0000,,at vinkel ABC er kongruent med vinkel DCB.
Dialogue: 0,0:07:09.39,0:07:12.89,Default,,0000,0000,0000,,.
Dialogue: 0,0:07:14.04,0:07:18.63,Default,,0000,0000,0000,,.
Dialogue: 0,0:07:18.64,0:07:22.32,Default,,0000,0000,0000,,Den her vinkelen er altså lik med den her vinkelen.
Dialogue: 0,0:07:22.33,0:07:25.48,Default,,0000,0000,0000,,Igjen kan de her være tilsvarende innvendige vinkler.
Dialogue: 0,0:07:25.49,0:07:27.46,Default,,0000,0000,0000,,Det her er en transversal,
Dialogue: 0,0:07:27.47,0:07:29.96,Default,,0000,0000,0000,,og her er 2 linjer, som vi ikke er helt sikre på er parallelle.
Dialogue: 0,0:07:29.97,0:07:33.11,Default,,0000,0000,0000,,Fordi de tilsvarende innvendige vinkler er kongruente,
Dialogue: 0,0:07:33.12,0:07:34.70,Default,,0000,0000,0000,,vet vi, at de er parallelle.
Dialogue: 0,0:07:34.71,0:07:36.96,Default,,0000,0000,0000,,.
Dialogue: 0,0:07:36.97,0:07:44.51,Default,,0000,0000,0000,,AC er altså parallell med BD.
Dialogue: 0,0:07:48.63,0:07:49.55,Default,,0000,0000,0000,,Vi er ferdige.
Dialogue: 0,0:07:49.56,0:07:51.43,Default,,0000,0000,0000,,.
Dialogue: 0,0:07:51.44,0:07:55.63,Default,,0000,0000,0000,,Vi har vist,
Dialogue: 0,0:07:55.64,0:07:57.44,Default,,0000,0000,0000,,at motstående sier i et parallellogram er like lange.
Dialogue: 0,0:07:57.45,0:08:00.13,Default,,0000,0000,0000,,Vi har også vist, at hvis de motstående sider er like lange,
Dialogue: 0,0:08:00.14,0:08:01.16,Default,,0000,0000,0000,,er det et parallellogram.
Dialogue: 0,0:08:01.17,0:08:03.54,Default,,0000,0000,0000,,Vi har altså bevist det begge veier.
Dialogue: 0,0:08:03.55,0:08:04.73,Default,,0000,0000,0000,,.
Dialogue: 0,0:08:04.74,0:08:06.88,Default,,0000,0000,0000,,.
Dialogue: 0,0:08:06.89,0:08:11.72,Default,,0000,0000,0000,,.
Dialogue: 0,0:08:11.73,0:08:15.75,Default,,0000,0000,0000,,.
Dialogue: 0,0:08:15.76,0:08:18.78,Default,,0000,0000,0000,,.
Dialogue: 0,0:08:18.79,0:08:20.05,Default,,0000,0000,0000,,.
Dialogue: 0,0:08:20.06,0:08:23.10,Default,,0000,0000,0000,,.
Dialogue: 0,0:08:23.11,0:08:26.68,Default,,0000,0000,0000,,.
Dialogue: 0,0:08:26.69,0:08:29.01,Default,,0000,0000,0000,,.