While we might look at ∆ABC initially, we only know one of the three angles so we can't use it alone to find ∠1. We could use the Angle Sum Theorem to find ∠BCD and then knowing that ∠1 is supplementary to that, but there's an even easier way.
Since ∠1 is the exterior angle to ∆BCD, and we already know the remote interior angles of that triangle, we can use the Exterior Angle Theorem. That will give us this equation.
m∠1 = m∠CBD + m∠CDB m∠1 = 17 + 21 m∠1 = 38
That's all we need.
Find the measure of ∠2.
When we first look at ∆CDE, we only have ∠2 and 31° labeled. However, ∠DCE and ∠1 are vertical angles, and you know what that means: they're as congruent as they'll ever be.
Since we already know that m∠1 = 38, we know that m∠DCE = 38 also. Now that we know two of the angles, we can find m∠2 using the Angle Sum Theorem.
We already know ∠1, but would it be possible to solve the problem if we didn't? Sure thang!
Take a look at ∆ABD. One angle is 21°, another is ∠3, and the top angle is 85° + 17°. (We can do that because of the Angle Addition Postulate.) So, in case we've forgotten that ∠1 is 38°, we can still solve this using the Angle Sum Theorem.