Lösung von Zusatzaufgabe 12.2P (WS 12 13): Unterschied zwischen den Versionen

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{| class="wikitable sortable"
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! !!
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|-
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| Voraussetzung || Dreieck ABC mit den Innenwinkeln <math>\alpha ,\beta ,\gamma</math>
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|-
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| Behauptung || <math>\left| \alpha  \right| +\left| \beta  \right| +\left| \gamma  \right| = 180</math>
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|}
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<br />
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{| class="wikitable sortable"
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!Beweisschritte!!Begründung
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|-
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|1. Wir konstruieren eine Gerade g, für die gilt g ll <math>\overline{AB}</math> ^ C<math>\in</math>g    ||  Parallelenaxiom, Vor.
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|-
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|2. D(mb,180)(A)=C ^ D(mb,180)(B)=B' ||1.), Def. Punktspiegelung, Def. Mittelpunkt
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|-
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|2.1 <math>B'C \equiv g</math>  ||1.),2.)
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|-
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|3. <math>\alpha \tilde {=}\alpha '</math> || Wechselwinkelsatz, 1.),2.),2.1), Eig. Punktspiegelung (winkeltreue), winkelmaßerhaltend
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|-
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|4. D(ma,180)(A)=A' ^D(ma,180)(B)=C || 1.), Def. Punktspiegelung, Def. Mittelpunkt
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|-
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|5. <math>\beta \tilde {=} \beta'</math> || 4.),2.1) Wechselwinkelsatz, Eig. Punktspiegelung (winkeltreue), winkelmaßerhaltend
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|-
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|6. <math>\left| \alpha'  \right| + \left| \beta' \right|+ \left| \gamma  \right|= 180</math> || 4.), 5.),Def. Nebenwinkel, Satz(Nebenwinkel sind supplementär)
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|-
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|7. <math>\left| \alpha  \right| + \left| \beta \right|+ \left| \gamma  \right|= 180</math> || 3.),5.),6.)
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|}
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<br />--[[Benutzer:TobiWan|TobiWan]] 00:37, 3. Feb. 2013 (CET)<br />
 
[[Kategorie:Einführung_P]]
 
[[Kategorie:Einführung_P]]

Version vom 3. Februar 2013, 00:37 Uhr

Beweisen Sie den Innenwinkelsatz für Dreiecke mit Hilfe zweier Punktspiegelungen.



Voraussetzung Dreieck ABC mit den Innenwinkeln \alpha ,\beta ,\gamma
Behauptung \left| \alpha \right| +\left| \beta \right| +\left| \gamma \right| = 180


Beweisschritte Begründung
1. Wir konstruieren eine Gerade g, für die gilt g ll \overline{AB} ^ C\ing Parallelenaxiom, Vor.
2. D(mb,180)(A)=C ^ D(mb,180)(B)=B' 1.), Def. Punktspiegelung, Def. Mittelpunkt
2.1 B'C \equiv g 1.),2.)
3. \alpha \tilde {=}\alpha ' Wechselwinkelsatz, 1.),2.),2.1), Eig. Punktspiegelung (winkeltreue), winkelmaßerhaltend
4. D(ma,180)(A)=A' ^D(ma,180)(B)=C 1.), Def. Punktspiegelung, Def. Mittelpunkt
5. \beta \tilde {=} \beta' 4.),2.1) Wechselwinkelsatz, Eig. Punktspiegelung (winkeltreue), winkelmaßerhaltend
6. \left| \alpha' \right| + \left| \beta' \right|+ \left| \gamma \right|= 180 4.), 5.),Def. Nebenwinkel, Satz(Nebenwinkel sind supplementär)
7. \left| \alpha \right| + \left| \beta \right|+ \left| \gamma \right|= 180 3.),5.),6.)


--TobiWan 00:37, 3. Feb. 2013 (CET)

Von „http://geometrie.zum.de/index.php?title=Lösung_von_Zusatzaufgabe_12.2P_(WS_12_13)&oldid=21323“