Before NG, the vowel changes to sound more like the AY (/eɪ/) diphthong. The middle part of the tongue lifts toward the roof of the mouth, then the front part arches up.

Americans pronounce angry as ANG-gree (/ˈæŋgri/). In "angry", the "a" vowel before NG shifts toward "ay" — sounding like "ay" as in "say", a distinctly American pattern — most prominent in Midwestern American English; other GenAm speakers may use a less raised vowel. This is called the Cat-Vowel Before NG, and it's why Americans sound more relaxed than the textbook. It comes out as ANG·gree. Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "Ignore the angry ghost in the grey garage".
Record yourself saying "angry" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 5 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Raise the back of your tongue to touch the soft palate. Add vocal cord vibration, then release.

Curl or bunch your tongue without letting the tip touch the roof of your mouth. Brace the sides of your tongue against your upper back teeth, and round your lips slightly.
Pull the corners of your lips back slightly. Arch the middle-front of your tongue high toward the roof of the mouth.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "angry", the "a" vowel before NG shifts toward "ay" — sounding like "ay" as in "say", a distinctly American pattern — most prominent in Midwestern American English; other GenAm speakers may use a less raised vowel. Vowel changes to sound like /eɪ/ ("ay" as in "say").
Stress falls on the first syllable, not the others. Stretch ANG — keep everything else short and quick.