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". The original statement is "if two angles are complementary, then the angles are acute". So the converse is "if two angles are acute, then the angles are complementary".

Step2: Analyze converse for counter - example

Acute angles are angles with measure less than 90°. Two acute angles like 30° and 40° have a sum of 70° (less than 90°) and are not complementary. But two acute angles like 45° and 45° are complementary. A counter - example is when you have two acute angles, say 20° and 30°, and the sum of the measures of the angles is 50° which is not 90°. So 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 the original statement is "if two angles are not complementary, then the angles are not acute".

Step4: Analyze inverse for counter - example

If two angles are not complementary, they could still be acute. For example, two angles of 30° and 40° are not complementary but are acute. A counter - example is two acute angles with sum not equal to 90°. 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 the original statement is "if you have two angles that are not acute, then the angles are not complementary".

Step6: Analyze contrapositive

If an angle is not acute (it is right or obtuse), then the sum of two such non - acute angles will be greater than or equal to 90° + 90°=180° or one non - acute and one acute will be greater than 90°. So they cannot be complementary (complementary angles sum to 90°). So the contrapositive is true.

Answer:

Converse: If two angles are acute, then the angles are complementary. Counterexample: You have two acute angles (e.g., 20° and 30°), and the sum of the measures of the angles is 50°≠90°. The converse is false.
Inverse: If two angles are not complementary, then the angles are not acute. Counterexample: You have two acute angles (e.g., 30° and 40°) and the sum of the measures of the angles is 70°≠90°. The inverse is false.
Contrapositive: If you have two angles that are not acute, then the angles are not complementary. The contrapositive is true.