[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.72,0:00:02.55,Default,,0000,0000,0000,,Her har vi et parallellogram.
Dialogue: 0,0:00:02.56,0:00:06.66,Default,,0000,0000,0000,,Vi vil bevise, at diagonalene halverer hverandre.
Dialogue: 0,0:00:06.67,0:00:10.04,Default,,0000,0000,0000,,Det er ikke bare diagonaler.
Dialogue: 0,0:00:10.05,0:00:12.46,Default,,0000,0000,0000,,Det er linjer, som krysser parallelle linjer.
Dialogue: 0,0:00:12.47,0:00:14.56,Default,,0000,0000,0000,,Derfor kan vi også kalle de transversaler.
Dialogue: 0,0:00:14.57,0:00:19.54,Default,,0000,0000,0000,,Vi kan se på DB her,
Dialogue: 0,0:00:19.55,0:00:21.89,Default,,0000,0000,0000,,at den krysser DC og AB.
Dialogue: 0,0:00:21.90,0:00:23.64,Default,,0000,0000,0000,,Vi vet, at det kalles parallellogrammer.
Dialogue: 0,0:00:23.65,0:00:24.96,Default,,0000,0000,0000,,Vi vet, at de er parallelle.
Dialogue: 0,0:00:24.97,0:00:25.99,Default,,0000,0000,0000,,Det er et parallellogram.
Dialogue: 0,0:00:26.00,0:00:28.64,Default,,0000,0000,0000,,Tilsvarende innvendige vinkler er kongruente.
Dialogue: 0,0:00:28.65,0:00:31.36,Default,,0000,0000,0000,,Den her vinkelen er altså lik med den her vinkelen.
Dialogue: 0,0:00:31.37,0:00:32.67,Default,,0000,0000,0000,,.
Dialogue: 0,0:00:32.68,0:00:34.03,Default,,0000,0000,0000,,Vi kaller punktet i midten for E.
Dialogue: 0,0:00:34.04,0:00:42.63,Default,,0000,0000,0000,,Vinkel ABE er altså kongruent med vinkel CDE,
Dialogue: 0,0:00:42.64,0:00:50.13,Default,,0000,0000,0000,,for de er tilsvarende innvendige vinkler
Dialogue: 0,0:00:50.14,0:00:52.13,Default,,0000,0000,0000,,dannet av en transversal, som krysser parallelle linjer.
Dialogue: 0,0:00:52.14,0:00:56.68,Default,,0000,0000,0000,,.
Dialogue: 0,0:00:56.69,0:01:00.84,Default,,0000,0000,0000,,Diagonalen AC, eller transversalen AC,
Dialogue: 0,0:01:00.85,0:01:02.52,Default,,0000,0000,0000,,kan vi si det samme om.
Dialogue: 0,0:01:02.73,0:01:04.47,Default,,0000,0000,0000,,Den krysser her og her.
Dialogue: 0,0:01:04.48,0:01:06.22,Default,,0000,0000,0000,,De her 2 linjene er parallelle.
Dialogue: 0,0:01:06.23,0:01:09.36,Default,,0000,0000,0000,,.
Dialogue: 0,0:01:09.37,0:01:12.74,Default,,0000,0000,0000,,.
Dialogue: 0,0:01:12.75,0:01:19.05,Default,,0000,0000,0000,,Vinkel DEC er altså kongruent med vinkel BAE
Dialogue: 0,0:01:24.78,0:01:27.15,Default,,0000,0000,0000,,på nøyaktig sammen måte.
Dialogue: 0,0:01:27.16,0:01:28.68,Default,,0000,0000,0000,,.
Dialogue: 0,0:01:28.69,0:01:31.58,Default,,0000,0000,0000,,Hvis vi ser på den øverste trekanten og den nederste trekanten,
Dialogue: 0,0:01:31.59,0:01:34.82,Default,,0000,0000,0000,,har vi et sett tilsvarende vinkler, som er kongruente.
Dialogue: 0,0:01:34.83,0:01:39.61,Default,,0000,0000,0000,,Siden her mellom er kongruent.
Dialogue: 0,0:01:39.62,0:01:41.22,Default,,0000,0000,0000,,.
Dialogue: 0,0:01:41.23,0:01:46.38,Default,,0000,0000,0000,,Vi har tidligere bevist,
Dialogue: 0,0:01:46.67,0:01:50.38,Default,,0000,0000,0000,,at motstående sider i parallellogrammer både er
Dialogue: 0,0:01:50.39,0:01:51.54,Default,,0000,0000,0000,,parallelle og kongruente.
Dialogue: 0,0:01:51.55,0:01:54.31,Default,,0000,0000,0000,,Den her siden er altså
Dialogue: 0,0:01:54.32,0:01:55.23,Default,,0000,0000,0000,,lik med den her siden.
Dialogue: 0,0:01:55.24,0:01:56.84,Default,,0000,0000,0000,,.
Dialogue: 0,0:01:56.85,0:01:59.76,Default,,0000,0000,0000,,Vi har 2 sett tilsvarende vinkler, som er kongruente.
Dialogue: 0,0:01:59.77,0:02:02.71,Default,,0000,0000,0000,,Vi har en side mellom, som er kongruent.
Dialogue: 0,0:02:02.72,0:02:04.74,Default,,0000,0000,0000,,Her har vi ytterligere et sett tilsvarende vinkler,
Dialogue: 0,0:02:04.75,0:02:05.77,Default,,0000,0000,0000,,som er kongruente.
Dialogue: 0,0:02:05.78,0:02:08.15,Default,,0000,0000,0000,,Den her trekanten er altså kongruent med den her trekanten
Dialogue: 0,0:02:08.16,0:02:10.32,Default,,0000,0000,0000,,på grunn av vinkel-side-vinkelkongruens.
Dialogue: 0,0:02:11.81,0:02:15.96,Default,,0000,0000,0000,,.
Dialogue: 0,0:02:15.97,0:02:17.46,Default,,0000,0000,0000,,.
Dialogue: 0,0:02:17.47,0:02:23.12,Default,,0000,0000,0000,,Trekant ABE er altså kongruent med
Dialogue: 0,0:02:23.13,0:02:29.97,Default,,0000,0000,0000,,trekant CDE med vinkel-side-vinkelkonguens.
Dialogue: 0,0:02:33.72,0:02:35.94,Default,,0000,0000,0000,,Hva forteller det oss?
Dialogue: 0,0:02:35.95,0:02:38.86,Default,,0000,0000,0000,,Hvis 2 trekanter er kongruente,
Dialogue: 0,0:02:38.87,0:02:41.37,Default,,0000,0000,0000,,er alle deres tilsvarende egenskaper og især
Dialogue: 0,0:02:41.38,0:02:42.62,Default,,0000,0000,0000,,tilsvarende sider kongruente.
Dialogue: 0,0:02:42.63,0:02:47.74,Default,,0000,0000,0000,,Side EC svarer til side EA.
Dialogue: 0,0:02:47.75,0:02:51.92,Default,,0000,0000,0000,,Vi kan også kalle de side AE
Dialogue: 0,0:02:55.24,0:02:59.47,Default,,0000,0000,0000,,og side CE.
Dialogue: 0,0:03:00.99,0:03:02.83,Default,,0000,0000,0000,,De er tilsvarende sider i kongruente trekanter.
Dialogue: 0,0:03:02.84,0:03:05.36,Default,,0000,0000,0000,,De er altså like lange.
Dialogue: 0,0:03:05.37,0:03:08.85,Default,,0000,0000,0000,,AE er lik med CE.
Dialogue: 0,0:03:08.86,0:03:12.32,Default,,0000,0000,0000,,Vi setter 2 streker for å markere det.
Dialogue: 0,0:03:18.21,0:03:24.32,Default,,0000,0000,0000,,BE må være lik med CE.
Dialogue: 0,0:03:25.95,0:03:29.45,Default,,0000,0000,0000,,Igjen er de tilsvarende sider i 2 kongruente trekanter.
Dialogue: 0,0:03:29.46,0:03:30.87,Default,,0000,0000,0000,,De må derfor være like lange.
Dialogue: 0,0:03:30.88,0:03:38.32,Default,,0000,0000,0000,,.
Dialogue: 0,0:03:38.33,0:03:43.00,Default,,0000,0000,0000,,BE er lik med DE.
Dialogue: 0,0:03:43.01,0:03:44.08,Default,,0000,0000,0000,,Vi er nå ferdige med beviset.
Dialogue: 0,0:03:44.09,0:03:48.78,Default,,0000,0000,0000,,Diagonalen DB deler diagonal AC opp i 2 deler,
Dialogue: 0,0:03:48.79,0:03:51.23,Default,,0000,0000,0000,,som er like lange.
Dialogue: 0,0:03:51.24,0:03:55.78,Default,,0000,0000,0000,,De 2 delene er like lange,
Dialogue: 0,0:03:55.79,0:03:58.07,Default,,0000,0000,0000,,og derfor halverer de hverandre.
Dialogue: 0,0:03:58.08,0:03:59.64,Default,,0000,0000,0000,,La oss prøve å bevise det omvendt.
Dialogue: 0,0:03:59.65,0:04:03.92,Default,,0000,0000,0000,,Vi skal bevise, at hvis vi i en firkant her 2 diagonaler,
Dialogue: 0,0:04:03.93,0:04:06.98,Default,,0000,0000,0000,,som halverer hverandre,
Dialogue: 0,0:04:06.99,0:04:08.81,Default,,0000,0000,0000,,er det et parallellogram.
Dialogue: 0,0:04:08.82,0:04:10.02,Default,,0000,0000,0000,,.
Dialogue: 0,0:04:10.03,0:04:12.01,Default,,0000,0000,0000,,Vi går ut fra,
Dialogue: 0,0:04:12.02,0:04:13.15,Default,,0000,0000,0000,,at de 2 diagonalene halverer hverandre.
Dialogue: 0,0:04:13.16,0:04:14.98,Default,,0000,0000,0000,,Den her må altså være lik med den her,
Dialogue: 0,0:04:14.99,0:04:17.36,Default,,0000,0000,0000,,og den her må være lik med den her.
Dialogue: 0,0:04:17.37,0:04:22.29,Default,,0000,0000,0000,,.
Dialogue: 0,0:04:22.30,0:04:25.16,Default,,0000,0000,0000,,.
Dialogue: 0,0:04:25.44,0:04:30.00,Default,,0000,0000,0000,,Vi skal huske, at den her vinkelen
Dialogue: 0,0:04:30.01,0:04:31.04,Default,,0000,0000,0000,,må være lik med den her vinkelen.
Dialogue: 0,0:04:31.05,0:04:33.73,Default,,0000,0000,0000,,De er toppvinkler.
Dialogue: 0,0:04:33.74,0:04:34.64,Default,,0000,0000,0000,,.
Dialogue: 0,0:04:34.65,0:04:43.58,Default,,0000,0000,0000,,Vinkel CED er lik med,
Dialogue: 0,0:04:43.59,0:04:52.39,Default,,0000,0000,0000,,det vil si kongruent med, vinkel BEA.
Dialogue: 0,0:04:52.40,0:04:55.20,Default,,0000,0000,0000,,Det forteller oss,
Dialogue: 0,0:04:55.21,0:04:57.81,Default,,0000,0000,0000,,at de 2 trekantene er kongruente, fordi de er tilsvarende sider.
Dialogue: 0,0:04:57.82,0:05:00.31,Default,,0000,0000,0000,,.
Dialogue: 0,0:05:00.32,0:05:03.81,Default,,0000,0000,0000,,Trekant AEB må altså være
Dialogue: 0,0:05:03.82,0:05:20.30,Default,,0000,0000,0000,,side-vinkel-sidekongruent med trekant DEC.
Dialogue: 0,0:05:20.31,0:05:28.17,Default,,0000,0000,0000,,.
Dialogue: 0,0:05:28.18,0:05:29.16,Default,,0000,0000,0000,,.
Dialogue: 0,0:05:29.17,0:05:31.76,Default,,0000,0000,0000,,Når 2 trekanter er kongruente, vet vi,
Dialogue: 0,0:05:31.77,0:05:34.22,Default,,0000,0000,0000,,at alle tilsvarende sider og vinkler er kongruente.
Dialogue: 0,0:05:34.23,0:05:44.58,Default,,0000,0000,0000,,Eksempelvis vet vi,
Dialogue: 0,0:05:44.59,0:05:48.36,Default,,0000,0000,0000,,at vinkel CDE er kongruent med vinkel BAE.
Dialogue: 0,0:05:55.65,0:06:05.79,Default,,0000,0000,0000,,Det er tilsvarende vinkler i kongruente trekanter.
Dialogue: 0,0:06:05.80,0:06:12.43,Default,,0000,0000,0000,,Det er en slags transversal til de her 2 linjene,
Dialogue: 0,0:06:12.44,0:06:16.57,Default,,0000,0000,0000,,som er parallelle, hvis de tilsvarende innvendige vinklene er kongruente.
Dialogue: 0,0:06:16.58,0:06:17.99,Default,,0000,0000,0000,,Det kan vi se, at de er.
Dialogue: 0,0:06:18.00,0:06:22.47,Default,,0000,0000,0000,,Det er tilsvarende innvendige vinkler,
Dialogue: 0,0:06:22.48,0:06:23.91,Default,,0000,0000,0000,,og de er kongruente.
Dialogue: 0,0:06:23.92,0:06:26.87,Default,,0000,0000,0000,,AB må derfor være parallell med CD.
Dialogue: 0,0:06:26.88,0:06:31.78,Default,,0000,0000,0000,,Vi tegner en pil for å markere,
Dialogue: 0,0:06:34.95,0:06:42.62,Default,,0000,0000,0000,,at vinkel AB er parallell med vinkel CD.
Dialogue: 0,0:06:42.80,0:06:46.11,Default,,0000,0000,0000,,.
Dialogue: 0,0:06:46.12,0:06:47.67,Default,,0000,0000,0000,,.
Dialogue: 0,0:06:47.68,0:06:50.30,Default,,0000,0000,0000,,Vi kan bruke helt samme metode til å vise,
Dialogue: 0,0:06:50.31,0:06:53.23,Default,,0000,0000,0000,,at de her 2 sidene er parallelle.
Dialogue: 0,0:06:53.24,0:06:55.64,Default,,0000,0000,0000,,.
Dialogue: 0,0:06:55.65,0:06:57.09,Default,,0000,0000,0000,,Vi behøver ikke nødvendigvis skrive det hele igjen.
Dialogue: 0,0:06:57.10,0:06:59.97,Default,,0000,0000,0000,,Det er det samme, vi skal gjøre.
Dialogue: 0,0:06:59.98,0:07:03.68,Default,,0000,0000,0000,,Den her vinkelen er kongruent med den her vinkelen.
Dialogue: 0,0:07:03.69,0:07:04.63,Default,,0000,0000,0000,,.
Dialogue: 0,0:07:04.64,0:07:06.93,Default,,0000,0000,0000,,Vi vet også,
Dialogue: 0,0:07:06.94,0:07:18.67,Default,,0000,0000,0000,,at vinkel AEC er kongruent med vinkel DEB.
Dialogue: 0,0:07:22.65,0:07:24.36,Default,,0000,0000,0000,,De er toppvinkler.
Dialogue: 0,0:07:26.98,0:07:29.06,Default,,0000,0000,0000,,.
Dialogue: 0,0:07:29.07,0:07:31.92,Default,,0000,0000,0000,,.
Dialogue: 0,0:07:31.93,0:07:35.26,Default,,0000,0000,0000,,Vi ser derfor, at trekant AEC må være
Dialogue: 0,0:07:35.27,0:07:38.27,Default,,0000,0000,0000,,side-vinkel-sidekongruent med trekant DEB.
Dialogue: 0,0:07:38.60,0:07:45.01,Default,,0000,0000,0000,,.
Dialogue: 0,0:07:45.02,0:07:50.89,Default,,0000,0000,0000,,.
Dialogue: 0,0:07:50.90,0:07:53.73,Default,,0000,0000,0000,,Vi vet, at tilsvarende vinkler er kongruente.
Dialogue: 0,0:07:53.74,0:07:58.68,Default,,0000,0000,0000,,For eksempel er vinkel CAE
Dialogue: 0,0:08:01.76,0:08:10.97,Default,,0000,0000,0000,,kongruent med vinkel BDE,
Dialogue: 0,0:08:10.98,0:08:13.51,Default,,0000,0000,0000,,fordi de er tilsvarende vinkler i kongruente trekanter.
Dialogue: 0,0:08:13.52,0:08:17.95,Default,,0000,0000,0000,,.
Dialogue: 0,0:08:18.13,0:08:25.94,Default,,0000,0000,0000,,CEA må være kongruent med BDE.
Dialogue: 0,0:08:28.05,0:08:30.10,Default,,0000,0000,0000,,Her har vi så en transversal.
Dialogue: 0,0:08:30.11,0:08:32.10,Default,,0000,0000,0000,,De tilsvarende innvendige vinklene er kongruente.
Dialogue: 0,0:08:32.11,0:08:34.69,Default,,0000,0000,0000,,De 2 linjene, som transversalen krysser,
Dialogue: 0,0:08:34.70,0:08:36.13,Default,,0000,0000,0000,,må være parallelle.
Dialogue: 0,0:08:36.14,0:08:39.23,Default,,0000,0000,0000,,De her er altså parallelle.
Dialogue: 0,0:08:39.24,0:08:44.44,Default,,0000,0000,0000,,AV er parallell med BD.
Dialogue: 0,0:08:45.49,0:08:47.97,Default,,0000,0000,0000,,.
Dialogue: 0,0:08:50.56,0:08:51.36,Default,,0000,0000,0000,,Nå er vi ferdige.
Dialogue: 0,0:08:51.37,0:08:53.97,Default,,0000,0000,0000,,.
Dialogue: 0,0:08:53.98,0:08:57.91,Default,,0000,0000,0000,,Når vi går ut fra, at diagonalen halverer hverandre,
Dialogue: 0,0:08:57.92,0:09:00.86,Default,,0000,0000,0000,,ender vi med å si, at de motstående sidene i firkanten er parallelle,
Dialogue: 0,0:09:00.87,0:09:04.69,Default,,0000,0000,0000,,og derfor er firkantene et parallellogram.