We performed a reference-free measurement of the transition energies of the 1s2p1P1→1s21S0 line in He-like argon, and of the 1s2s22p1P1→1s22s21S0 line in Be-like argon ions. The highly charged ions were produced in the plasma of an electron-cyclotron resonance ion source. Both energy measurements were performed with an accuracy better than 3 parts in 106, using a double flat-crystal spectrometer, without reference to any theoretical or experimental energy. The 1s2s22p1P1→1s22s21S0 transition measurement is the first reference-free measurement for this core-excited transition. The 1s2p1P1→1s21S0 transition measurement confirms recent measurement performed at the Heidelberg electron-beam ion trap. The width measurement in the He-like transition provides test of a purely radiative decay calculation. In the case of the Be-like argon transition, the width results from the sum of a radiative channel and three main Auger channels. We also performed multiconfiguration Dirac-Fock calculations of transition energies and rates and have done an extensive comparison with theory and other experimental data. For both measurements reported here, we find agreement with the most recent theoretical calculations within the combined theoretical and experimental uncertainties.