what ground cover can i use on sloped terrain in my yard. it is very hard to mow and i would like to not do dangerous areas with my mower.?

If my wife sees that I suggested this, she may clout me with a stick: English ivy.

NOTE: Engilsh ivy is very invasive and will climb trees, to the detriment of the trees. If your sloped area has no trees and you mow all the way around it, you will keep the ivy contained and need not worry. That is the ONLY circumstance under which I would actually plant English ivy.

Other good choices:

Blue rug juniper

Vetch

Phlox

Vinca

There are other good choices out there but that should get you started.

Please don't plant English ivy or Japanese honeysuckle. They are very invasive, not just on your own property but will spread to natural areas by birds and damage your local ecosystems. I would recommend looking into some no-mow grasses like buffalo grass or Carex pensylvanica. They grow a little longer than typical lawn grasses and flop over. I like the look. They also don't require mowing or irrigation once they are established. Another great option depending on your climate is an aggressive fern like ostrich fern or hay-scented fern. They spread quickly and will cover the ground.

You need to tell us where your slope is.

In my area lantana is excellent. Blooms attracts butterflies.

And is invasive in Australia.

Avoid English ivy. Very invasive, kills trees and all the understory plants in the areas that it invades.

Avoid vinca, terribly invasive.

Avoid winter creeper euonymus. Invasive menace.

Why not sow a nice patch of clover?

Or better yet... Why not let the native plants grow... And then enjoy the butterflies. Who says that you need to plant invasive exotics anyway?

shore junper

blue rug juniper

Creeping vinca, ajuga, honeysuckle or euonymous. If shady try pachysandra or ajuga also.