Question

2 - 4 skills practice deductive reasoning
determine whether the stated conclusion is valid based on the given information. if not, write invalid. explain your reasoning.

1. given: if the sum of the measures of two angles is 180, then the angles are supplementary. (mangle a + mangle b) is 180.

conclusion: (angle a) and (angle b) are supplementary.

1. given: if the sum of the measures of two angles is 90, then the angles are complementary. (mangle abc) is 45 and (mangle def) is 48.

conclusion: (angle abc) and (angle def) are complementary.

1. given: if the sum of the measures of two angles is 180, then the angles are supplementary. (angle1) and (angle2) are a linear - pair.

conclusion: (angle1) and (angle2) are supplementary.
use the law of syllogism to draw a valid conclusion from each set of statements, if possible. if no valid conclusion can be drawn write no valid conclusion and explain your reasoning.

1. if two angles are complementary, then the sum of their measures is 90. if the sum of the measures of two angles is 90, then both of the angles are acute.
2. if the heat wave continues, then air - conditioning will be used more frequently. if air - conditioning is used more frequently, then energy costs will be higher.
3. if it is tuesday, then marla tutors chemistry. if marla tutors chemistry, then she arrives home at 4 p.m.
4. if a marine animal is a starfish, then it lives in the intertidal zone of the ocean. the intertidal zone is the least stable of the ocean zones.

Explanation:

Step1: Recall the definition of supplementary angles

If the sum of the measures of two angles is 180, they are supplementary. Given \(m\angle A + m\angle B=180\), by definition \(\angle A\) and \(\angle B\) are supplementary. So the conclusion in 1 is valid.

Step2: Recall the definition of complementary angles

Complementary angles have a sum of 90 degrees. Given \(m\angle ABC = 45\) and \(m\angle DEF=48\), \(m\angle ABC+m\angle DEF=45 + 48=93
eq90\). So the conclusion in 2 that \(\angle ABC\) and \(\angle DEF\) are complementary is invalid.

Step3: Recall the property of linear - pair and supplementary angles

A linear - pair of angles has a sum of 180 degrees. If \(\angle1\) and \(\angle2\) are a linear pair, then by definition they are supplementary. So the conclusion in 3 is valid.

Step4: Apply the Law of Syllogism

The Law of Syllogism states that if \(p
ightarrow q\) and \(q
ightarrow r\), then \(p
ightarrow r\). In 4, if two angles are complementary (\(p\)), then the sum of their measures is 90 (\(q\)), and if the sum of the measures of two angles is 90 (\(q\)), then both angles are acute (\(r\)). So if two angles are complementary, then both angles are acute, and the conclusion is valid.

Step5: Apply the Law of Syllogism

In 5, if the heat - wave continues (\(p\)), then air - conditioning will be used more frequently (\(q\)), and if air - conditioning is used more frequently (\(q\)), then energy costs will be higher (\(r\)). By the Law of Syllogism, if the heat - wave continues, then energy costs will be higher.

Step6: Apply the Law of Syllogism

In 6, if it is Tuesday (\(p\)), then Marla tutors chemistry (\(q\)), and if Marla tutors chemistry (\(q\)), then she arrives home at 4 PM (\(r\)). By the Law of Syllogism, if it is Tuesday, then Marla arrives home at 4 PM.

Step7: Analyze the statements in 7

The first statement is "If a marine animal is a starfish (\(p\)), then it lives in the intertidal zone of the ocean (\(q\))". The second statement is "The intertidal zone is the least stable of the ocean zones (\(r\))". There is no way to form a new conditional statement using the Law of Syllogism here, so no valid conclusion can be drawn.

Answer:

1. Valid
2. Invalid
3. Valid
4. Valid
5. If the heat - wave continues, then energy costs will be higher.
6. If it is Tuesday, then Marla arrives home at 4 PM.
7. No valid conclusion.