We can match each idea to its definition by saying that claim is the ideas of a speaker (first box) while evidence is facts and examples (second box).
What is claim?
Claim can be defined as the opinion or point of view expressed by an author concerning a certain topic, issue, or subject. The author then presents arguments that will defend and support his claim. It is important to add evidence to those arguments, if possible.
What is evidence?
Evidence, in its turn, is anything an author can provide as support for his claim. Evidence includes:
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