1. Identify the molecule and geometry around nitrogen

Triethyl amine is $(C_2H_5)_3N$.

The nitrogen atom is bonded to three carbon atoms and has one lone pair. So, around nitrogen there are 4 electron pairs:

• 3 bond pairs
• 1 lone pair

Hence, the electron pair geometry is approximately tetrahedral.

2. Effect of lone pair on bond angle

For a perfect tetrahedral arrangement, the bond angle is $109.5^\circ$

But a lone pair repels more strongly than bond pairs, so it compresses the bond angle between the three $C-N$ bonds.

Thus, the $C-N-C$ bond angle in a trialkyl amine is slightly less than $109.5^\circ$.

3. Known bond angle in gaseous triethyl amine

In gaseous triethyl amine, due to pyramidal $sp^3$ nitrogen geometry, the $C-N-C$ bond angle is approximately $108^\circ$

4. Final answer

Therefore, the required bond angle is $\boxed{108}$

degree.