anoroclupita2008
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Bobby and Elaine each write a proof for the statement [tex]\(m \angle DCB = 95^\circ\)[/tex].

Given: [tex]\(m \angle ACD = 85^\circ\)[/tex]
Prove: [tex]\(m \angle DCB = 95^\circ\)[/tex]

Bobby's Proof:
By the linear pair theorem, [tex]\(\angle ACD\)[/tex] is supplementary to [tex]\(\angle DCB\)[/tex]. This means that [tex]\(m \angle ACD + m \angle DCB = 180^\circ\)[/tex]. Since [tex]\(m \angle ACD = 85^\circ\)[/tex], the substitution property of equality implies that [tex]\(85^\circ + m \angle DCB = 180^\circ\)[/tex]. Applying the subtraction property of equality, [tex]\(m \angle DCB = 95^\circ\)[/tex].

Elaine's Proof:
Suppose [tex]\(m \angle DCB \neq 95^\circ\)[/tex]. By the linear pair theorem, [tex]\(\angle ACD\)[/tex] is supplementary to [tex]\(\angle DCB\)[/tex]. This means that [tex]\(m \angle ACD + m \angle DCB = 180^\circ\)[/tex]. Using the substitution property of equality, this means that [tex]\(m \angle ACD + 95^\circ \neq 180^\circ\)[/tex]. Applying the subtraction property of equality, [tex]\(m \angle ACD \neq 85^\circ\)[/tex]. Since this contradicts what is given, then [tex]\(m \angle DCB = 95^\circ\)[/tex].

What type of proofs did they use?

Bobby used [tex]\(\square\)[/tex] because [tex]\(\square\)[/tex].
Elaine used [tex]\(\square\)[/tex] because [tex]\(\square\)[/tex].

Sagot :

GinnyAnswer
Certainly, let's analyze the proofs one by one and identify the type of proof each method describes.

### Bobby's Proof:
Bobby’s proof involves directly showing that [tex]\(m \angle DCB = 95^\circ\)[/tex] from the given facts.

1. Bobby starts with the given fact that [tex]\(\angle ACD\)[/tex] is supplementary to [tex]\(\angle DCB\)[/tex], which means [tex]\(m \angle ACD + m \angle DCB = 180^\circ\)[/tex].
2. He then substitutes the given value [tex]\(m \angle ACD = 85^\circ\)[/tex] into the equation.
3. This leads to [tex]\(85^\circ + m \angle DCB = 180^\circ\)[/tex].
4. By subtracting [tex]\(85^\circ\)[/tex] from both sides, Bobby concludes that [tex]\(m \angle DCB = 95^\circ\)[/tex].

This is a direct proof because Bobby directly finds the value of [tex]\(m \angle DCB\)[/tex] from the given information and the properties of supplementary angles.

### Elaine's Proof:
Elaine’s proof involves assuming the negation of the statement and demonstrating that it leads to a contradiction.

1. Elaine begins by assuming the opposite of what needs to be proven: [tex]\(m \angle DCB \neq 95^\circ\)[/tex].
2. By the linear pair theorem, [tex]\(\angle ACD\)[/tex] is supplementary to [tex]\(\angle DCB\)[/tex], so [tex]\(m \angle ACD + m \angle DCB = 180^\circ\)[/tex].
3. Substituting the given [tex]\(m \angle ACD = 85^\circ\)[/tex] into the equation gives us [tex]\(85^\circ + m \angle DCB = 180^\circ\)[/tex].
4. From the assumption [tex]\(m \angle DCB \neq 95^\circ\)[/tex], it would imply that [tex]\(85^\circ + 95^\circ \neq 180^\circ\)[/tex], which contradicts the supplementary angle theorem since [tex]\(85^\circ + 95^\circ = 180^\circ\)[/tex].
5. This contradiction implies that the assumption was wrong, thereby [tex]\(m \angle DCB\)[/tex] must be [tex]\(95^\circ\)[/tex].

This is a proof by contradiction because Elaine assumes the negation of the statement to be proven and shows that this assumption leads to a contradiction.

Therefore:

Bobby used direct proof because he provided a direct argument from the given information to the conclusion.

Elaine used proof by contradiction because she assumed the negation of what she wanted to prove and showed that it led to a contradiction.
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