Question

1. geometry if two angles are complementary, then the angles are acute.

Explanation:

Let's solve problem 20 (GEOMETRY) step by step.

Step 1: Identify the original conditional statement

The original statement is: "If two angles are complementary, then the angles are acute."

Step 2: Recall the definitions of converse, inverse, contrapositive

• Converse: Switch the hypothesis and conclusion. So, "If the angles are acute, then two angles are complementary."
• Inverse: Negate both hypothesis and conclusion. So, "If two angles are not complementary, then the angles are not acute."
• Contrapositive: Negate and switch hypothesis and conclusion. So, "If the angles are not acute, then two angles are not complementary."

Step 3: Determine the truth value of the original statement

Complementary angles sum to \(90^\circ\). An acute angle is less than \(90^\circ\). If two angles are complementary, each must be less than \(90^\circ\) (since their sum is \(90^\circ\)), so they are acute. So the original statement is true.

Step 4: Determine the truth value of the converse

Take two acute angles, say \(30^\circ\) and \(40^\circ\). Their sum is \(70^\circ\), not \(90^\circ\), so they are not complementary. Thus, the converse is false. A counterexample is two acute angles that don't sum to \(90^\circ\) (e.g., \(30^\circ\) and \(40^\circ\)).

Step 5: Determine the truth value of the inverse

The inverse is "If two angles are not complementary, then the angles are not acute." But we can have two non - complementary angles that are acute (like the \(30^\circ\) and \(40^\circ\) example above). So the inverse is false. A counterexample is two acute, non - complementary angles.

Step 6: Determine the truth value of the contrapositive

The contrapositive is "If the angles are not acute, then two angles are not complementary." If an angle is not acute, it is \(\geq90^\circ\). If two angles are not acute, their sum is \(\geq90^\circ + 90^\circ=180^\circ\), so they can't be complementary (since complementary angles sum to \(90^\circ\)). So the contrapositive is true.

Answer:

• Original Statement: "If two angles are complementary, then the angles are acute." - True.
• Converse: "If the angles are acute, then two angles are complementary." - False. Counterexample: \(30^\circ\) and \(40^\circ\) (acute but not complementary).
• Inverse: "If two angles are not complementary, then the angles are not acute." - False. Counterexample: \(30^\circ\) and \(40^\circ\) (not complementary but acute).
• Contrapositive: "If the angles are not acute, then two angles are not complementary." - True.