Question

1. What is the negation of the following: "If this triangle has two 45 degree angles then it is a right triangle."

Answer 1

Answer:

Let  

• p : this triangle has two 45 degree angles  
• q : it is a right triangle

Step-by-step explanation:

Denote

• r : if this triangle has two 45 degree angles then it is a right triangle

Then, r can be written  as follows

• $p \rightarrow q$

Since $(p \rightarrow q) \iff (\neg p \lor q)$ is a tautology, you get

• $\neg(p\rightarrow q)\iff\neg(\neg p \lor q)$  

and given that  

• $\neg(p\rightarrow q)\iff(p \land \neg q) (\mbox{double negation and Morgan's Lows})$

it follows that the negation of r is

• " this triangle has two 45 degree angles and it is not a right triangle".

Then, the correct answer is C.