Question

1. which law of logic is being used to draw the conclusion?

if two angles are vertical angles, then they are congruent. if two angles are congruent, then their measure is the same. angles z
and y are vertical angles.
conclusion: angles z and y have the same measure.
a. law of detachment
b. law of syllogism

1. all reptiles have scales. crocodiles are reptiles. therefore, crocodiles have scales.

do the statements above demonstrate inductive or deductive reasoning?
a. inductive
b. deductive

1. determine the next term in this pattern:

3, 6, 12, 24, ...
a. 48
b. 36
c. 102
d. 96

Explanation:

Question 8

The Law of Syllogism states that if \(p
ightarrow q\) and \(q
ightarrow r\), then \(p
ightarrow r\). Here, \(p\) is “two angles are vertical angles”, \(q\) is “they are congruent”, and \(r\) is “their measure is the same”. We have \(p
ightarrow q\) and \(q
ightarrow r\), and then using the fact that \(p\) (angles \(Z\) and \(Y\) are vertical angles) to get \(p
ightarrow r\) (angles \(Z\) and \(Y\) have the same measure). The Law of Detachment is for the form \(p
ightarrow q\) and \(p\) (to get \(q\)), but here we have a chain of implications.

Deductive reasoning starts with a general statement (all reptiles have scales - a general rule) and applies it to a specific case (crocodiles are reptiles - a specific instance) to draw a specific conclusion (crocodiles have scales). Inductive reasoning would be making a general rule from specific observations.

Step1: Identify the pattern

The pattern is a geometric sequence. The common ratio \(r\) can be found by dividing a term by its previous term. For example, \(r=\frac{6}{3} = 2\), \(\frac{12}{6}=2\), \(\frac{24}{12}=2\).

Step2: Find the next term

The formula for the \(n\)th term of a geometric sequence is \(a_{n}=a_{1}r^{n - 1}\). Here \(a_{1} = 3\), \(r = 2\), and we want the 5th term (\(n=5\)). Using the formula \(a_{5}=3\times2^{5 - 1}=3\times2^{4}\).

$$a_{5}=3\times16=48$$

Answer:

B. Law of Syllogism

Question 9