Question

mixed exercises construct an argument draw a valid conclusion from the given statements, if possible. then state whether your conclusion was drawn using the law of detachment or the law of syllogism. if no valid conclusion can be drawn, write no valid conclusion. justify your argument. 18. given: if a figure is a square, then all the sides are congruent. figure abcd is a square. 19. given: if two angles are complementary, the sum of the measures of the angles is 90°. ∠1 and ∠2 are complementary angles. 20. given: ballet dancers like classical music. if you like classical music, then you enjoy the opera.

Explanation:

18.

Step1: Identify the form

The first - given statement is a conditional statement "If a figure is a square, then all the sides are congruent" (let \(p\): a figure is a square, \(q\): all the sides are congruent, so \(p
ightarrow q\)). The second - given statement is \(p\) (Figure \(ABCD\) is a square).

Step2: Apply the Law of Detachment

The Law of Detachment states that if \(p
ightarrow q\) is true and \(p\) is true, then \(q\) is true.

Step1: Identify the form

The first - given statement is a conditional statement "If two angles are complementary, the sum of the measures of the angles is \(90^{\circ}\)" (let \(p\): two angles are complementary, \(q\): the sum of the measures of the angles is \(90^{\circ}\), so \(p
ightarrow q\)). The second - given statement is \(p\) (\(\angle1\) and \(\angle2\) are complementary angles).

Step2: Apply the Law of Detachment

By the Law of Detachment, if \(p
ightarrow q\) is true and \(p\) is true, then \(q\) is true.

Step1: Identify the forms

Let \(p\): a person is a ballet dancer, \(q\): a person likes classical music, \(r\): a person enjoys the opera. The first statement is \(p
ightarrow q\), and the second statement is \(q
ightarrow r\).

Step2: Apply the Law of Syllogism

The Law of Syllogism states that if \(p
ightarrow q\) and \(q
ightarrow r\), then \(p
ightarrow r\).

Answer:

Conclusion: All the sides of figure \(ABCD\) are congruent. The conclusion was drawn using the Law of Detachment.

19.