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I've been told, If $p$ then $q$ we must also have if $\sim p$ then $\sim q$. I don't quite get this.

Let $p \implies q$ : if a and b are vertical angles, they are congruent.

The contrapostive should mean that if two angles are not vertical they are not congruent right?

But I don't think that's right, Let $a$ and $b$ be supplementary ,linear, angles both measuring $90$ degrees. They are not vertical angles but they are congruent. Is this contradicting what I've been told? Am I missing something?

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    You've got it backwards. The contrapositive is "if $p$ then $q$" implies "if not $q$ then not $p$". If it's raining hard, then it's wet outside. It's not wet outside, so it must not be raining hard. – 2017-02-27

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Whoever told you that if you have $p \Rightarrow q$ means you have $\neg p \Rightarrow \neg q$ was wrong. For example consider square implies rectangle but not square doesn't imply not rectangle. What is true is that $p \Rightarrow q$ and $ \neg q \Rightarrow \neg p$ are the same. They are the contrapositive of each other. You can verify this by writing out truth tables for the implications if you want.

Here are the truth tables:

$$\begin{array}{ccc}P&Q&P \Rightarrow Q&\neg Q&\neg P&\neg Q \Rightarrow \neg P\\T&T&T&F&F&T\\T&F&F&T&F&F\\F&T&T&F&T&T\\ F&F&T&T&T&T\end{array}$$

We can also verify this identity by recognizing that $P \Rightarrow Q$ means $\neg P \vee Q$. So $\neg Q \Rightarrow \neg P$ means $\neg \neg Q \vee\neg P$ which is the same as $Q \vee \neg P$ so the two statements are equivalent.

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    Thanks, I think I didn't pay attention to the switching it part when I was told in my speech in debate class. I was probably told what you said but wasn't paying close attention. Anyways, now I know so thanks. I will accept in a bit. – 2017-02-27
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    No problem. I added the truth tables to help you visualize it a bit better. You can also verify it through boolean arithmetic and rewriting what implication means. – 2017-02-27