Question

peter’s proof:
by the linear pair theorem, ∠1 is supplementary to ∠2. so, ( mangle1 + mangle2 = 180^circ ). since ( angle2 cong angle3 ), then ( angle2 = angle3 ). applying the transitive property of equality, ( mangle1 + mangle3 = 180^circ ), which means ( angle1 ) is supplementary to ( angle3 )

vivian’s proof:
suppose ( angle1 ) is not supplementary to ( angle3 ). so, ( mangle1 + mangle3
eq 180^circ ). by the linear pair theorem, ( angle1 ) is supplementary to ( angle2 ). by the definition of supplementary angles, ( mangle1 + mangle2 = 180^circ ). applying the transitive property, ( mangle1 + mangle3
eq mangle1 + mangle2 ). by the subtraction property of equality, this implies that ( mangle3
eq mangle2 ). by definition of congruence, ( mangle3 cong mangle2 ). however, ( mangle3 cong mangle2 ) contradicts the given.
what type of proofs did they use?

Explanation:

Peter's proof starts with known theorems (linear pair theorem) and given congruence (\(\angle 2\cong\angle 3\)) to derive that \(\angle 1\) is supplementary to \(\angle 3\), which is a direct proof (starts with premises and uses logical steps to reach conclusion). Vivian's proof assumes the opposite of the conclusion (\(\angle 1\) is not supplementary to \(\angle 3\)) and shows this leads to a contradiction (contradicts \(\angle 2\cong\angle 3\)), so it's a proof by contradiction (indirect proof).

Answer:

Peter used a direct proof (starts from known facts and deduces the conclusion). Vivian used a proof by contradiction (indirect proof, assumes the negation of the conclusion and shows it leads to a contradiction).