Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

Americans pronounce damages as DA-muh-juhz (/ˈdæmədʒəz/). In "damages", the "a" vowel before M or N raises and fronts toward [eə] — the tongue pulls up and forward, breaking the vowel into a tense glide as it anticipates the nasal. This is called the Cat-Vowel Before M/N, and it's why Americans sound more relaxed than the textbook. It comes out as DA·muh·juhz. Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "He is liable for the damages caused by his negligence" or "The landlord sued the tenant for unpaid rent and damages" — more examples below.
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3 syllables, 7 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

The tongue relaxes down in the back and the corners of the lips relax before the consonant. This adds a schwa-like 'uh' relaxation after the /æ/. Think of it as 'relaxing out of the vowel' — it is no longer a pure /æ/ sound.

Click any sentence to see the full breakdown — every link, every reduction, every flap-T.
The textbook way isn't wrong — it's just not how anyone actually says it.
In "damages", the "a" vowel before M or N raises and fronts toward [eə] — the tongue pulls up and forward, breaking the vowel into a tense glide as it anticipates the nasal. The "/æ/" vowel raises and fronts before M or N — tongue pulls up and forward, producing a tense [eə] glide (between /e/ and /ə/). Not a pure /æ/.
Stress falls on the first syllable, not the others. Stretch DA — keep everything else short and quick.
Don't pronounce the first syllable too fully. The unstressed syllable reduces to a schwa — the lazy "uh" sound — in casual speech.