You may know one or both of the words in the title above: der Mut means "the courage" and mutig means "courageous" or "brave." However, there are many nouns, verbs, adjectives, and adverbs that contain these words as roots, some of which have nothing to do with courage. Let's have a look.

As you can see, nouns ending in -mut may refer to a mood, emotional state, or feeling:

Hochmut kommt vor dem Fall.

Pride comes before the fall. 

Caption 24, Eva erklärt - Sprichwörter

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Was bei vielen Autofahrern für Freude sorgt,

What causes happiness among many drivers,

stiftet Unmut bei den Grünen.

causes discontent among the Greens [Green Party].

Caption 19, Deutsche Autobahnen - Geschwindigkeitsbegrenzungen

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Die Queen residierte hier bereits und Michael Jackson

The Queen resided here already and Michael Jackson

hielt im Übermut sein Baby aus einem Hotelfenster.

held his baby boisterously out of a hotel window.

Captions 24-25, Berlin - Hotel Adlon feiert 15 Jahre Neueröffnung

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...in spektakulärer Anmut auf die Wasseroberfläche zu klatschen.

...to smack against the surface of the water with spectacular grace.

Caption 8, Evolution - Meeresbewohner

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Notice above that der Hochmut, der Übermut, and der Unmut are masculine, while die Anmut is feminine. You can also look up der Wagemut, der Missmut, die Sanftmut, and die Schwermut. For der Demut, you may more often see the related adjective (note the umlaut!):

Es war jedenfalls demütigend genug.

In any case, it was humiliating enough.

Caption 20, Lerchenberg - Das Wunder

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It's additionally good to know the adjectives ermutigend and entmutigend. These respectively mean "encouraging" and "discouraging."

When it comes to verbs, you may have seen zumuten, vermuten, and anmuten. These verbs each have a number of possible meanings.

Man vermutet, hier ist irgendwas in der Lüftung vom Willy-Brandt-Haus.

One suspects there is something in the ventilation here in the Willy-Brandt-Haus.

Caption 75, heute-show - Die männliche Merkel hat Erinnerungslücken

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Traditionen sind was Schönes

Traditions are something nice

und können für Nichteingeweihte mitunter ganz schön skurril anmuten.

and can occasionally appear quite bizarre to the outsider.

Captions 1-2, Barfuß unter Schafen - Schäferwettrennen

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Petrus muss Dortmund-Fan sein, anders ist kaum zu erklären,

Saint Peter must be a Dortmund fan, there's almost no other explanation

was der Himmel dem FC Bayern und seinen Fans zugemutet hat.

for what the heavens had in store for FC Bayern and its fans.

Captions 1-2, FC Bayern München - Triple-Feier im Dauerregen

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Further Learning
Look up these words using your favorite online dictionary or on Yabla German to understand the full extent of their meanings. Keep in mind that there are other nouns ending in -mut that are unrelated to emotional states, such as das Mammut or der Wermut.