Lingerie (pronounced lë-zhree in French, with 'ë' representing a nasalised i) is a word that has been absorbed into English and the pronunciation modified (lon-zher-ay). Go to Forvo to hear native French speakers saying it. Lingerie in both languages means the same thing.
However, brassiere is a false friend. A brassière is what I would call a singlet, and I think only used in reference to baby clothes here. French bras are called soutiens-gorge ('throat supports'!). They don't have cups, but bonnets ('bonnets'). French sizing, however, is a perfectly sensible system - centimetres around the body just under the breast, ie your band measurement, not some randomly assigned number, and cup sizes are indicated by letters of the alphabet, A being the smallest volume.
à droite, un soutien-gorge à armatures (underwire bra) et un slip.
*Push-up bras are often referred to as les balconnets in France (for the technicalities see this post from The Stage Door, in French). They give you a noticeable décolleté (cleavage). My understanding is that the English decolletage, meaning cleavage, is a false friend. The French word décolletage isn't used much and when it is, only refers to a deep or wide neckline of a garment, and not directly to the view it affords of the body beneath. Ken may be able to clarify.