Question

which answers are examples of the law of detachment? select each correct answer if an object is a square, then it is a rhombus. if it is a rhombus, then it is a quadrilateral. therefore, if an object is a square, then it is a quadrilateral jonathans cars are all white. sarah is driving one of jonathans cars. therefore, sarah is driving a white car if two angles are vertical angles, then they have the same measure. angle a and angle b are vertical angles. so, the measure of angle a is equal to the measure of angle b. if it snows tomorrow, then my dentist appointment will be canceled. if my dentist appointment is canceled, then i will clean under my bed. therefore, if it snows tomorrow, then i will clean under my bed

Explanation:

Step1: Recall the Law of Detachment

The Law of Detachment states that if \(p
ightarrow q\) (if \(p\) then \(q\)) is a true conditional statement and \(p\) (the hypothesis) is true, then \(q\) (the conclusion) is true.

Step2: Analyze each option

• First option:

It uses the Law of Syllogism (\(p
ightarrow q\) and \(q
ightarrow r\) implies \(p
ightarrow r\)) where \(p\) = “an object is a square”, \(q\) = “it is a rhombus”, \(r\) = “it is a quadrilateral”. So, it is not an example of the Law of Detachment.

• Second option:

Let \(p\) = “Sarah is driving one of Jonathan’s cars” and \(q\) = “Sarah is driving a white car”. The statement “Jonathan’s cars are all white” is \(p
ightarrow q\). Since \(p\) (Sarah is driving one of Jonathan’s cars) is true, by the Law of Detachment, \(q\) (Sarah is driving a white car) is true.

• Third option:

Let \(p\) = “Angle \(A\) and angle \(B\) are vertical angles” and \(q\) = “the measure of angle \(A\) is equal to the measure of angle \(B\)”. The statement “If two angles are vertical angles, then they have the same measure” is \(p
ightarrow q\). Since \(p\) (Angle \(A\) and angle \(B\) are vertical angles) is true, by the Law of Detachment, \(q\) (the measure of angle \(A\) is equal to the measure of angle \(B\)) is true.

• Fourth option:

It uses the Law of Syllogism (\(p
ightarrow q\) and \(q
ightarrow r\) implies \(p
ightarrow r\)) where \(p\) = “it snows tomorrow”, \(q\) = “my dentist appointment will be canceled”, \(r\) = “I will clean under my bed”. So, it is not an example of the Law of Detachment.

Answer:

Jonathan’s cars are all white. Sarah is driving one of Jonathan’s cars. Therefore, Sarah is driving a white car.
If two angles are vertical angles, then they have the same measure. Angle \(A\) and angle \(B\) are vertical angles. So, the measure of angle \(A\) is equal to the measure of angle \(B\).