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Bobby and Elaine each write a proof for the statement [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][tex]$m \angle ACD = 85^{\circ}$[/tex][/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][tex]$m \angle DCB \neq 95^{\circ}$[/tex][/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][tex]$m \angle ACD + 95^{\circ} \neq 180^{\circ}$[/tex][/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].
Bobby used a direct proof because he started with given information and used logical steps to arrive at the conclusion.
Elaine used proof by contradiction because she assumed the negation of the statement to show that it leads to a contradiction with the given information.
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