5，One of the things that fascinates us most about cats is the popular belief that they have nine lives.

本句中，that 引导同位语从句, 修饰the popular belief(通俗的信念)

e.g. One of the things that fascinates us most about planes is the popular fact that they are safe and fast.

6．Apparently, there is a good deal of truth in this idea.

ruth,不可数名词，a good deal of

apparently 显然，显而易见，语气不确定，相当于It seems that...

7．A cat's ability to survive falls is based on fact.

survive falls 在跌落时存活，fall 做名词

be based on:基于事实基础之上

base sth. on: 把某事加强在什么基础之上

e.g. We base the theory on the fact. 我们把理论建立在事实基础上。

e.g. The theory is based on fact. 这一理论是基于事实之上的。重庆新概念英语

8．Recently the New York Animal medical Centre made a study of 132 cats over a period of five months.

make a study of sth:对什么作出研究

over a period of five months