You may not be running Ubuntu, but here https://help.ubuntu.com/community/UEFI the Ubuntu folks describe how to install using EFI.
What you need is to create a bootable USB stick for EFI with UNetBootin and then copy the install iso onto it. Also you might need to disable Secureboot in your EFI Configuration.
A word on Windows on just a 80GB partition: It is really small and you propably are only able to install very few programs until you run out of space. As for linux it might be okay with 80GB, but still not much space to work with.
UPDATE: Creating a boot stick to install linux
To create a bootable USB Stick usable for EFI install on Windows run
diskpart and enter the following commands. Be sure to select the USB stick in select disk where you want to place the setup files for linux.
- list disk
- select disk <#>
- create partition primary
- select partition 1
- format quick fs=fat32
Then mount the ISO of the linuxdistribution you want to install and just copy the contents. You then should be able to boot from the USB Stick. If not disable secure boot in EFI. To setup secureboot for Grub2 to be able to boot Windows have a look at this article http://www.boerngen-schmidt.de/2015/04/gentoo-and-windows-8-1-secure-boot-setup/