Question

for the given conditional statement, determine which of the following option(s) has a truth value of true. select all that apply. if a polygon is regular, then it has congruent angles and congruent sides. conditional inverse converse contrapositive

Explanation:

Step 1: Analyze the Conditional Statement

The given conditional statement is "If a polygon is regular, then it has congruent angles and congruent sides." By definition, a regular polygon is a polygon that is equiangular (all angles congruent) and equilateral (all sides congruent). So this conditional statement is true.

Step 2: Analyze the Inverse

The inverse of a conditional statement "If \( p \), then \( q \)" is "If not \( p \), then not \( q \)". For our statement, \( p \): "a polygon is regular", \( q \): "it has congruent angles and congruent sides". The inverse would be "If a polygon is not regular, then it does not have congruent angles and congruent sides". But there are non - regular polygons (e.g., a rectangle which has congruent angles but not all sides congruent, or a rhombus which has congruent sides but not all angles congruent, and also some polygons can have both congruent angles and sides by coincidence but not be regular) that can have congruent angles or congruent sides. So the inverse is false.

Step 3: Analyze the Converse

The converse of "If \( p \), then \( q \)" is "If \( q \), then \( p \)". So the converse here is "If a polygon has congruent angles and congruent sides, then it is regular". By the definition of a regular polygon, a polygon with congruent angles and congruent sides is regular. So the converse is true.

Step 4: Analyze the Contrapositive

The contrapositive of "If \( p \), then \( q \)" is "If not \( q \), then not \( p \)". The contrapositive of our statement is "If a polygon does not have congruent angles and congruent sides, then it is not regular". Since a regular polygon must have both congruent angles and sides, if a polygon lacks either (or both), it can't be regular. So the contrapositive is true. Also, we know that a conditional statement and its contrapositive always have the same truth value. Since the original conditional is true, the contrapositive is also true.

Answer:

conditional, converse, contrapositive