Question

texes mathematics 4-8 (115)

1. given the following statements:

in a given \\(\delta abc\\), \\(m\angle c = 90^\circ\\), \\(m\angle a = 44^\circ\\), and \\(m\angle b = 46^\circ\\). in a given \\(\delta def\\), \\(m\angle f = 90^\circ\\), \\(m\angle d = 30^\circ\\), and \\(m\angle e = 60^\circ\\).

is the claim, \the sum of the acute angles of any right triangle must be \\(90^\circ\\)\ deductive or inductive reasoning?

select only one:
a. deductive
b. inductive

Explanation:

🆕 New Concept Discovered: Inductive vs. Deductive Reasoning
Using specific examples to form a general rule vs. using general rules to prove a specific case.

Step 1: Analyze the given statements

We are given specific measurements for two right triangles:

• In \(\triangle ABC\), \(m\angle C = 90^\circ\), \(m\angle A = 44^\circ\), and \(m\angle B = 46^\circ\).
• The sum of the acute angles is: \(44^\circ + 46^\circ = 90^\circ\).
• In \(\triangle DEF\), \(m\angle F = 90^\circ\), \(m\angle D = 30^\circ\), and \(m\angle E = 60^\circ\).
• The sum of the acute angles is: \(30^\circ + 60^\circ = 90^\circ\).

Step 2: Determine the type of reasoning used to make the claim

The claim is: "The sum of the acute angles of any right triangle must be \(90^\circ\)."

• Inductive reasoning starts with specific observations or examples (like \(\triangle ABC\) and \(\triangle DEF\)) and arrives at a general conclusion or rule.
• Deductive reasoning starts with general rules, facts, or definitions and applies them to a specific situation to prove it must be true.

Because the claim is a general rule formulated by observing specific examples, the reasoning used is inductive.

Answer:

B. Inductive