What is the bond angle of the C-N-H bond angle in (CH3)2NH?

This compound is drawn like so, with the #"CNH"# atoms highlighted:

This is called dimethylamine; amine, because of the #"NH"#, and dimethyl because of the two #"CH"_3# groups on the nitrogen.

(You might also call this N,N-dimethylamine to emphasize the locations of the methyl groups.)

We can see the lone pair of electrons on the nitrogen, which immediately means that one of the #"CNH"# bond angles will not be "standard", so to speak.

(This compound is NOT trigonal planar, by the way.)

You can see that it has four electron groups: two methyls, one hydrogen, and one lone pair. That corresponds to a tetrahedral electron geometry, and thus a trigonal pyramidal molecular geometry.

As a result, we can expect the bond angle to be close to #109.5^@#.

However, due to the lone pair of electrons, which take up quite a bit of space when they aren't bonding, the molecule "crunches" up a little bit, and the angle becomes LESS than #109.5^@#. We can see this better when we redraw the compound like so:

Looking up the actual bond angle, it is about #color(blue)(108.9^@)#. You can look it up yourself as well here:

https://tutor.hix.ai

Simply enter the molecular formula, then choose the three atoms you want for your bond angle (four if it is a dihedral/torsion angle).