this is the reason 【why】he was late for school? this is the

the reason why ...意为 什么 的原因；the reason that/which 后跟修饰reason的定语从句,比如the reason he gave us 意为 他给我们的理由.
如果表示的是“做什么事的原因”之类,就用why；如果是“他给的原因”之类,就用that/which.
体会一下下面两句话的含义.
this is the reason 【that】he gave me yesterday.这是他昨天给我的（某件事情的）原因.
this is the reason 【why】he gave me yesterday.这是他为什么昨天给我的原因.