ClF5 has 42 electrons to account for. 10 of them are in bonding pairs of electrons. 30 more are in lone pairs on F atoms. That accounts for 40 electrons. Two electrons are left on Cl as a lone pair. Since there are six electron pairs on the central atom, then VSEPR theory tells us that the electron pair geometry is octahedral and with 5 bonding pairs and 1 lone pair, the molecular geometry is square pyramidal. The bond angles will be slightly less than 90 degrees because of the additional repulsion from the delocalized lone pair.
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Old science guy has misplaced two electrons, giving him the wrong electron pair geometry.