Question

uses the graph to provide the answers to the following questions.

conditional:_____________________
truth value:___ explain:___________________
converse:_____________________
truth value:___ explain:___________________
inverse:_____________________
truth value: ___ explain:___________________
contrapositive: _____________________
truth value: ___ explain:___________________
biconditional:_____________________

(second part with angle a and b)

conditional:_____________________
converse:_____________________
inverse:_____________________
contrapositive:_____________________
biconditional:_____________________

Answer:

To solve this, we'll analyze the first angle diagram (right angle with a ray inside, forming two acute angles) and the second (linear pair with angles \(a\) and \(b\)). Let's start with the first diagram (right angle, e.g., \(\angle AOB = 90^\circ\), ray \(OC\) inside, so \(\angle AOC\) and \(\angle COB\) are acute).

First Diagram (Right Angle with Interior Ray)

Let’s define a conditional statement. For example, let the hypothesis be "an angle is the larger angle formed by the ray" and conclusion "it is acute" (or adjust based on typical angle logic).

1. Conditional:

Statement: If an angle is formed by the ray and one axis (e.g., the vertical axis), then it is acute.
Truth Value: True (since the total angle is \(90^\circ\), any sub - angle is \(< 90^\circ\), so acute).
Explain: The original angle is a right angle (\(90^\circ\)). A ray inside it creates two angles, both smaller than \(90^\circ\) (acute).

2. Converse:

Statement: If an angle is acute, then it is formed by the ray and one axis.
Truth Value: False.
Explain: There are many acute angles not related to this specific ray/axis setup (e.g., a \(30^\circ\) angle in a different diagram).

3. Inverse:

Statement: If an angle is not formed by the ray and one axis, then it is not acute.
Truth Value: False.
Explain: Angles not in this setup (e.g., a \(45^\circ\) angle elsewhere) can still be acute.

4. Contrapositive:

Statement: If an angle is not acute, then it is not formed by the ray and one axis.
Truth Value: True.
Explain: The ray - axis angles are acute (\(< 90^\circ\)). So a non - acute angle (\(\geq 90^\circ\)) can’t be one of these.

5. Biconditional:

Statement: An angle is formed by the ray and one axis if and only if it is acute.
Truth Value: False.
Explain: The converse is false, so the biconditional (which requires both conditional and converse to be true) is false.

Second Diagram (Linear Pair: \(a + b=180^\circ\))

Let’s define a conditional. For example, hypothesis: "an angle is \(a\)", conclusion: "it is supplementary to \(b\)".

1. Conditional:

Statement: If an angle is \(a\), then it is supplementary to \(b\).
Truth Value: True.
Explain: \(a + b = 180^\circ\) (linear pair), so \(a\) and \(b\) are supplementary.

2. Converse:

Statement: If an angle is supplementary to \(b\), then it is \(a\).
Truth Value: False.
Explain: Any angle \(x\) with \(x + b=180^\circ\) is supplementary to \(b\), not just \(a\) (e.g., \(x = 180^\circ - b\) could be another angle in a different context).

3. Inverse:

Statement: If an angle is not \(a\), then it is not supplementary to \(b\).
Truth Value: False.
Explain: Angles other than \(a\) (e.g., \(180^\circ - b\) from another diagram) can still be supplementary to \(b\).

4. Contrapositive:

Statement: If an angle is not supplementary to \(b\), then it is not \(a\).
Truth Value: True.
Explain: If an angle is not supplementary to \(b\) (i.e., \(x + b
eq180^\circ\)), it can’t be \(a\) (since \(a + b = 180^\circ\)).

5. Biconditional:

Statement: An angle is \(a\) if and only if it is supplementary to \(b\).
Truth Value: False.
Explain: The converse is false, so the biconditional fails.

Final Answer (Example for First Diagram - Conditional)

For the first diagram’s Conditional:

• Statement: If an angle is formed by the ray and the vertical axis, then it is acute.
• Truth Value: True
• Explain: The original angle is \(90^\circ\), so sub - angles are \(< 90^\circ\) (acute).

(Adjust based on the specific diagram’s angle labels; the key is applying logical statement rules to angle relationships.)