Spiders produce different kinds of silk for different purposes. A spider will use dragline silk, which is the strongest kind of silk, to make the spokes of the web and any part that is intended to bear the weight of the spider. This silk isn't sticky and the spider can safely move over it to different parts of the web. The spiral connections of the web are sticky and are intended to snare prey.
Spiders will only touch the web with the tips of their feet to minimize the chances of becoming stuck. They also spend time grooming their feet, and may secrete substances onto their feet that prevent sticking to the web.
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