Question

write the converse, inverse, and contrapositive of the true conditional statement. determine whether the related conditional is true or false. if a statement is false, then find a counterexample. geometry if two angles are complementary, then the angles are acute. converse: if select choice, then select choice. counterexample: you have two acute angles, and the sum of the measures of the angles select choice 90°. the converse is select choice. inverse: if two angles are not complementary, then the angles select choice acute. counterexample: you have two acute angles, and the sum of the measures of the angles select choice 90°. the inverse is select choice. contrapositive: if you have two angles that are not acute, then the angles select choice complementary. the contrapositive is select choice.

Explanation:

Step1: Define converse

The converse of a conditional statement "if p then q" is "if q then p". For the statement "If two angles are complementary, then the angles are acute", the converse is "If two angles are acute, then the angles are complementary".

Step2: Analyze converse truth - value

An acute angle has a measure between 0° and 90°. For example, two acute angles of 30° and 40° have a sum of 70°≠90°. So, if you have two acute angles, the sum of the measures of the angles may not be 90°. The converse is false.

Step3: Define inverse

The inverse of "if p then q" is "if not p then not q". So the inverse of "If two angles are complementary, then the angles are acute" is "If two angles are not complementary, then the angles are not acute".

Step4: Analyze inverse truth - value

Counter - example: Consider two acute angles of 30° and 50°. They are not complementary (since 30 + 50=80≠90), but they are acute. So the inverse is false.

Step5: Define contrapositive

The contrapositive of "if p then q" is "if not q then not p". So the contrapositive of "If two angles are complementary, then the angles are acute" is "If two angles are not acute, then the angles are not complementary".

Step6: Analyze contrapositive truth - value

If an angle is not acute (i.e., it is a right angle (90°) or an obtuse angle (greater than 90°)), then the sum of two such non - acute angles will be greater than or equal to 90°. So they cannot be complementary (since complementary angles sum to 90°). The contrapositive is true.

Answer:

Converse: If two angles are acute, then the angles are complementary. False. Counterexample: Two acute angles with measures 30° and 40° (sum is 70°≠90°).
Inverse: If two angles are not complementary, then the angles are not acute. False. Counterexample: Acute angles of 30° and 50° (not complementary but acute).
Contrapositive: If two angles are not acute, then the angles are not complementary. True.