Altitudes line AD and line BE of triangle ABC intersect at H. If angle BAC = 46 degrees and angle ABC = 71 degrees, then what is angle AHB?

117 ^°

Just ' follow your nose '.

You know angles A and B, so work out angle C.

You can now work out angles CAH and CBE.

That allows you to work out angles HAB and HBA.

Knowing those allows you to work out AHB.

Job done.

Here's a pic :

We don't need as much work as the guest has laid out.....

First find angle ACB  = 180 - BAC  - CBA  = 180  - 180  - 46 - 71 =  63°

EHDC forms a quadrilateral whose interior angles sum to 360°

So   angle EHD  = 360 -  90 - 90 - 63  = 117°

And AHB  is a vertical angle to EHD =  117°

Thanks so much everyone!