In German, there are several different ways to talk about avoiding something or someone. 

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The verb that is used most often is probably vermeiden. You can use it to describe avoiding a situation or something you have to do. 

So vermeidest du am leichtesten eine peinliche Situation,

That's the easiest way for you to avoid an embarrassing situation

Caption 62, Deutsch mit Eylin: Tischsitten

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Der Schreiadler vermeidet die Konfrontation.

The lesser spotted eagle avoids confrontation.

Caption 16, Expeditionen ins Tierreich: Die Feldberger Seen

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Habe das einmal sieben, habe ich lang vermieden.

For a long time, I avoided the seven times table. 

Caption 83, Nena: Das 1x1 mit den Hits von NENA

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The verb meiden is similar, but generally used to describe staying away from something or someone, rather than an active effort to avoid doing something. 

Selbst Murmeltiere meiden diese Gegend.

Even marmots avoid this area.

Caption 17, Die letzten Paradiese - Schätze der Natur: Südtirol 

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The verb ausweichen is used when it comes to physically avoiding something in the sense of dodging it. 

Wildwasserkajaks sind extrem wendig, um den vielen Hindernissen im Fluss rasch ausweichen zu können.

Whitewater kayaks are extremely maneuverable so that they can quickly dodge the many obstacles in the river.

Captions 41-42, Die letzten Paradiese: Die Schönheit der Alpen 2

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When it comes to avoiding a person, there is also the phrase (jemandem) aus dem Weg gehen. 

Ich geh dir aus dem Weg, will dich ignorieren.

I avoid you, I want to ignore you.

Caption 7, LOTTE: Schau mich nicht so an

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Further Learning
Use each of these verbs in a sentence, or look up more examples with vermeiden on Yabla German.