Question

fill in the blanks to write the contrapositive of the following statement and complete its proof.
if a polygon has more than three sides, then it is not a triangle.
true false is not conditional statement has more than is converse has exactly
contrapositive: if a polygon □ a triangle, then the polygon □ three sides. since a triangle has exactly three sides, the contrapositive is □. since the contrapositive is □, the □ must be true.

Explanation:

Step1: Recall contrapositive definition

The contrapositive of a conditional statement "If \( p \), then \( q \)" is "If not \( q \), then not \( p \)". The original statement is: "If a polygon has more than three sides (\( p \)), then it is not a triangle (\( q \))". So not \( q \) is "a polygon is a triangle", and not \( p \) is "the polygon has exactly three sides".

Step2: Fill first blank

For the contrapositive "If a polygon \(\underline{\quad}\) a triangle", from not \( q \), we use "is" (since \( q \) was "is not a triangle", so not \( q \) is "is a triangle").

Step3: Fill second blank

For "then the polygon \(\underline{\quad}\) three sides", from not \( p \), \( p \) was "has more than three sides", so not \( p \) is "has exactly three sides".

Step4: Determine truth of contrapositive

Since a triangle has exactly three sides, the contrapositive "If a polygon is a triangle, then the polygon has exactly three sides" is true.

Step5: Relate to original statement

Since the contrapositive is true, the original conditional statement must be true (because a conditional statement and its contrapositive have the same truth value).

Answer:

Contrapositive: If a polygon \(\boldsymbol{\text{is}}\) a triangle, then the polygon \(\boldsymbol{\text{has exactly}}\) three sides. Since a triangle has exactly three sides, the contrapositive is \(\boldsymbol{\text{true}}\). Since the contrapositive is \(\boldsymbol{\text{true}}\), the \(\boldsymbol{\text{conditional statement}}\) must be true.