Question

select all statements that could be the first step of an indirect proof of the conditional below. if m∠2 = 110, then m∠1 = 70. a. if m∠2 = 110, then m∠1 = 70. b. if m∠1 ≠ 70, then m∠2 ≠ 110. c. assume if m∠1 ≠ 70, then m∠2 ≠ 110. d. assume if m∠2 ≠ 110, then m∠1 ≠ 70. e. assume if m∠2 = 70, then m∠1 = 110.

Explanation:

Step1: Recall indirect - proof concept

In an indirect proof of a conditional statement "If \(p\), then \(q\)" (\(p
ightarrow q\)), the first step is to assume the negation of the conclusion while keeping the hypothesis. The given conditional is "If \(m\angle2 = 110\), then \(m\angle1=70\)" where \(p:m\angle2 = 110\) and \(q:m\angle1 = 70\). The first step of an indirect proof is to assume \(p\) is true and \(q\) is false, i.e., assume \(m\angle2 = 110\) and \(m\angle1
eq70\). Another way to think about it is in terms of the contra - positive. The contra - positive of a conditional \(p
ightarrow q\) is \(
eg q
ightarrow
eg p\).

Step2: Analyze each option

• Option A is the original conditional, not the first step of an indirect proof.
• Option B is the contra - positive of the original conditional, not the first step of an indirect proof.
• Option C: Assuming \(m\angle1

eq70\) (negation of the conclusion) while considering the relationship with \(m\angle2\) in a way that is consistent with the start of an indirect proof.

• Option D is not the correct first step as it negates the hypothesis first.
• Option E is an incorrect assumption as it changes the values of the angles in an unrelated way to the indirect - proof process.

Answer:

C. Assume if \(m\angle1
eq70\), then \(m\angle2
eq110\)