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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][tex]$m \angle DCB = 95^{\circ}$[/tex][/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].

What type of proofs did they use?

Bobby used [tex]\square[/tex] because

Elaine used [tex]\square[/tex] because