The biologically active components of the methanol extracts of R. mucronata were identified using GC/MS. The anticancer effects of each methanol extract from the leaves and stem were evaluated against cancer and non-cancer cell lines. The MTT assay was used in order to evaluate cell viability, and the IC50 and the selectivity indices were calculated in relation to a positive control (doxorubicin). The results showed that 11 and 8 different chemical compounds were found in the methanol extracts from the leaves and stems of R. mucronata, respectively. The active constituents of R. mucronata leaves and stems had anticancer effects against colon cancer (CaCo-2), with IC50 levels of 127 ± 4 μg/mL and 107 ± 6 μg/mL, respectively, and on breast cancer (MCF-), with IC50 levels of 158 ± 10 μg/mL and 138 ± 4 μg/mL, respectively. These were both greater than their effects on prostate cancer (PC-3), for which they showed IC50 levels of 480 ± 14 μg/mL and 294 ± 3 μg/mL, respectively. However, the anticancer effect of the stems on lung cancer (A549) (IC50 = 155 ± 10 μg/mL) was greater than that of the leaves (IC50 = 376 ± 9 μg/mL) in comparison with doxorubicin. Neither the stems nor the leaves of R. mucronata showed any cytotoxicity against normal cells (WI-38), with the IC50 being 932 ± 30 μg/mL for the leaves and 629 ± 3 μg/mL for the stems.