The world is in danger.【正确】
The world is dangerous【错误】
in danger是说明the world处于“危险之中”,并不是说the world本身是“危险的”
dangerous是形容词,主语一般为人或动物.如：
He is a dangerous man.(他是个危险人物）=The man is dangerous.
Bears are one of the dangerous animals.(熊是危险的动物之一）=Bears are dangerous.
形容词主要用来修饰名词,表示事物的特征.dangerous显然不是the world的本身特征.