The proton-unbound argon and chlorine isotopes have been studied by measuring trajectories of their decay-in-flight products by using a tracking technique with microstrip detectors. The proton (1p) and two-proton (2p) emission processes have been detected in the measured angular correlations "heavy-fragment"+p and "heavy-fragment"+p+p, respectively. The ground states of the previously unknown isotopes Cl30 and Cl28 have been observed for the first time, providing the 1p-separation energies Sp of -0.48(2) and -1.60(8), MeV, respectively. The relevant systematics of 1p- and 2p-separation energies have been studied theoretically in the core+p and core+p+p cluster models. The first-time observed excited states of Ar31 allow one to infer the 2p-separation energy S2p of 6(34) keV for its ground state. The first-time observed state in Ar29 with S2p=-5.50(18) MeV can be identified as either a ground state or an excited state according to different systematics.