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 you have two acute angles, then the angles are complementary. inverse: if two angles are not complementary, then the angles are not acute. contrapositive: if you have two angles that are not acute, then the angles are not complementary. you have two acute angles, and the sum of the measures of the angles is 90°. the converse is select choice. inverse: if two angles are not complementary, then the angles are not acute. the inverse is select choice. contrapositive: if you have two angles that are not acute, then the angles are not complementary. the contrapositive is select choice. counterexample: you have two acute angles, and the sum of the measures of the angles is not 90°. the converse is select choice. counterexample: you have two angles that are not acute. counterexample

Explanation:

Step1: Define converse, inverse and contrapositive

The original conditional statement is "If two angles are complementary, then the angles are acute". The converse switches the hypothesis and conclusion, the inverse negates both the hypothesis and conclusion, and the contrapositive both negates and switches them.

Step2: Find the converse

The converse of "If \(p\), then \(q\)" is "If \(q\), then \(p\)". So the converse of "If two angles are complementary, then the angles are acute" is "If you have two acute angles, then the angles are complementary". This is false. A counter - example is two acute angles of \(30^{\circ}\) and \(40^{\circ}\), their sum is \(70^{\circ}
eq90^{\circ}\).

Step3: Find the 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". This is false. A counter - example is an angle of \(30^{\circ}\) and an angle of \(100^{\circ}\), they are not complementary and one is acute.

Step4: Find the 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 you have two angles that are not acute, then the angles are not complementary". This is true because non - acute angles (right or obtuse) cannot add up to \(90^{\circ}\).

Answer:

Converse: If you have two acute angles, then the angles are complementary. False, counter - example: two acute angles \(30^{\circ}\) and \(40^{\circ}\).
Inverse: If two angles are not complementary, then the angles are not acute. False, counter - example: an angle of \(30^{\circ}\) and an angle of \(100^{\circ}\).
Contrapositive: If you have two angles that are not acute, then the angles are not complementary. True.