We investigate the magnetic penetration depth lambda in superconducting Ba1-xKxFe2As2 (T-c similar or equal to 32 K) with muon-spin rotation (mu SR) and angle-resolved photoemission (ARPES). Using mu SR, we find the penetration-depth anisotropy gamma(lambda)=lambda(c)/lambda(ab) and the second-critical-field anisotropy gamma(Hc2) to show an opposite T evolution below T-c. This dichotomy resembles the situation in the two-gap superconductor MgB2. A two-gap scenario is also suggested by an inflection point in the in-plane penetration depth lambda(ab) around 7 K. The complementarity of mu SR and ARPES allows us to pinpoint the values of the two gaps and to arrive to a remarkable agreement between the two techniques concerning the full T evolution of lambda(ab). This provides further support for the described scenario and establishes ARPES as a tool to assess macroscopic properties of the superconducting condensate.