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:

∠efg ≅ ∠efg
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:

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The negation of a statement "The grapes are purple" is simply "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 logical negation just flips the truth - value of the original statement without adding new information about other colors.

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 equal quantities.

Deductive reasoning starts with a general rule (the formula for the sum of interior angles of a polygon \(S=180(n - 2)\)) and then applies it to a specific case (a triangle where \(n = 3\)). Inductive reasoning would involve making a generalization from specific examples. Here, facts (the formula and the definition of a triangle) are used in a top - down (general to specific) approach which is characteristic of deductive reasoning.

Answer:

A. The grapes are not purple

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