PLEASE ANSWER THE QUESTION WRITTEN IN DESCRIPTION

Question

 triangle has exactly three sides. Prove the following conditional statement by proving the contrapositive.

If a polygon has more than three sides, then it is not a triangle.

1. true
2. is not
3. conditional statement
4. has more than
5. is
6. converse
7. has exactly
8. false

Fill in the blanks to write the contrapositive.

If a polygon -------- a triangle, then the polygon--------  three sides.

Fill in the blanks to complete the proof.

Since a triangle has exactly three sides, the contrapositive is ----------.

Since the contrapositive is ------------- , the------------  must be true.

Answer 1

Conditional: if p, then q

p: "A polygon has more than 3 sides."

q: "It (the polygon) is not a triangle."

Contrapositive: if not q, then not p

not q: "It (the polygon) is a triangle."

not p: "A polygon does not have more than 3 sides." which also means "A polygon has 3 sides or less."

Given that the shape in the question is a polygon, the fewest number of sides any polygon can have is 3. This puts an additional constraint on "not p". Since a polygon cannot have less than three sides, "not p" becomes...

not p: "A polygon has 3 sides."

Next, we put the contrapositive into words.

IF not p, THEN not q.

IF a polygon has 3 sides, THEN it is a triangle. ✓

This conditional, which is the contrapositive of the original conditional, is true. If the contrapositive is true, then the original conditional is also true.

Now, I will attempt to fill in the blanks (eye-roll) although I've already explained it in a way that actually makes it easy to understand. But what is geometry without proofs?! Interesting!

If a polygon is a triangle, then the polygon has exactly three sides.

Since a triangle has exactly three sides, the contrapositive is true.

Since the contrapositive is true, the conditional must be true.