Question

1. using the statement: \the grapes are purple\, write the negation of the statement:

a. the grapes are not purple
b. the grapes are green
c. the grapes are green or purple
d. the grapes are not green

1. which property is illustrated by the statement:

\\( \angle e f g \cong \angle e f g \\)
a. reflexive property
b. transitive property
c. symmetric property
d. addition property

1. decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. explain.

the sum of the measures of the interior angles of a polygon is \\( 180(n-2) \\), where \\( n \\) is the number of sides.
a triangle is a polygon with three sides.
so, the sum of the interior angles of a triangle is \\( 180(3-2)=180 \\).
a. inductive reasoning, a pattern is used
b. inductive reasoning, facts are used
c. deductive reasoning, a pattern is used
d. deductive reasoning, facts are used

Explanation:

Question 4

The negation of a statement simply means the opposite. If the original statement is "The grapes are purple", then its negation is "The grapes are not purple". Options B, C, and D introduce other colors (green) or a disjunction which is not the direct logical negation.

The reflexive property states that any geometric figure (in this case, an angle \(\angle EFG\)) is congruent to itself (\(\angle EFG\cong\angle EFG\)). The transitive property is about if \(a = b\) and \(b = c\) then \(a = c\). The symmetric property is if \(a=b\) then \(b = a\). The addition property is related to adding quantities in equations.

Deductive reasoning starts with general facts (the formula for the sum of interior angles of a polygon \(180(n - 2)\) is a general formula, and a triangle is a specific case of a polygon with \(n=3\)). Inductive reasoning would involve looking for a pattern from specific examples. Here, we use a general formula (a fact) and apply it to a specific case (a triangle).

Answer:

A. The grapes are not purple

Question 5