Determine whether you can use the Law of Syllogism to reach a valid conclusion from each set of statements.

1. If a dog eats Superdog Dog Food, he will be happy.
Rover is happy.

2. If an angle is supplementary to an obtuse angle, then it is acute.
If an angle is acute, then its measurement is less than 90.

3. If the measure of angle A is less than 90, then angle A is acute.
If angle A is acute, then angle A is congruent to angle B.

Let's take the first one.

You can make a syllogism if you have two statements like:

(1) If A, then B _and_
(2) Rover is A
(3)_then_ you can say that Rover is B

Now:
(1) If a dog eats Superdog Dog Food, he will be happy.
(2) Rover is happy.

What is "A" here? it's "eats Superdog Dog Food". Does (2) say anything about "eats Superdog Dog Food"? No. So you don't have a syllogism. Rover might or might not eat Superdog Dog Food; we don't know.

So there is no syllogism here.

Now try the other two.

If an angle is a right angle, then the measure of the angle is 90. If two lines are perpendicular, then they form a right angle.

Let's determine whether we can use the Law of Syllogism to reach a valid conclusion from each set of statements:

1. If a dog eats Superdog Dog Food, he will be happy.
Rover is happy.

To use the Law of Syllogism, we need two conditional statements. In this case, we only have one conditional statement ("If a dog eats Superdog Dog Food, he will be happy") and an unrelated statement ("Rover is happy"). Therefore, we cannot use the Law of Syllogism to reach a valid conclusion.

2. If an angle is supplementary to an obtuse angle, then it is acute.
If an angle is acute, then its measurement is less than 90.

Here, we have two conditional statements. First, if an angle is supplementary to an obtuse angle, it is acute. Second, if an angle is acute, its measurement is less than 90. We can apply the Law of Syllogism, combining these two statements:
If an angle is supplementary to an obtuse angle, then its measurement is less than 90.

Therefore, we can use the Law of Syllogism to reach the valid conclusion that "if an angle is supplementary to an obtuse angle, its measurement is less than 90."

3. If the measure of angle A is less than 90, then angle A is acute.
If angle A is acute, then angle A is congruent to angle B.

Similar to the previous example, we have two conditional statements. First, if the measure of angle A is less than 90, then angle A is acute. Second, if angle A is acute, then angle A is congruent to angle B. Again, we can apply the Law of Syllogism and combine these two statements:
If the measure of angle A is less than 90, then angle A is congruent to angle B.

Therefore, we can use the Law of Syllogism to reach the valid conclusion that "if the measure of angle A is less than 90, then angle A is congruent to angle B."

To determine whether you can use the Law of Syllogism to reach a valid conclusion from each set of statements, we need to check if the statements follow the form of the Law of Syllogism:

The Law of Syllogism states that if the following two statements are true:
1. If p, then q.
2. If q, then r.

Then we can conclude:
3. If p, then r.

Let's analyze each set of statements:

1. If a dog eats Superdog Dog Food, he will be happy.
Rover is happy.

In this case, we have:
p: Rover eats Superdog Dog Food.
q: If Rover eats Superdog Dog Food, he will be happy.
r: Rover is happy.

The first statement does not provide any information about Rover eating Superdog Dog Food. Thus, we cannot use the Law of Syllogism to reach a valid conclusion.

2. If an angle is supplementary to an obtuse angle, then it is acute.
If an angle is acute, then its measurement is less than 90.

In this case, we have:
p: An angle is supplementary to an obtuse angle.
q: If an angle is supplementary to an obtuse angle, then it is acute.
r: If an angle is acute, then its measurement is less than 90.

The first statement gives us p → q.
The second statement gives us q → r.

Now, we can use the Law of Syllogism:
From p → q and q → r, we can conclude:
If an angle is supplementary to an obtuse angle, then its measurement is less than 90.
Therefore, we can use the Law of Syllogism for this set of statements.

3. If the measure of angle A is less than 90, then angle A is acute.
If angle A is acute, then angle A is congruent to angle B.

In this case, we have:
p: The measure of angle A is less than 90.
q: If the measure of angle A is less than 90, then angle A is acute.
r: If angle A is acute, then angle A is congruent to angle B.

The first statement gives us p → q.
The second statement gives us q → r.

Now, we can use the Law of Syllogism:
From p → q and q → r, we can conclude:
If the measure of angle A is less than 90, then angle A is congruent to angle B.
Therefore, we can use the Law of Syllogism for this set of statements.

In summary, we can use the Law of Syllogism in sets 2 and 3 to reach valid conclusions, but not in set 1.