Question

peter and vivian each wrote a proof for the statement: if ∠2 ≅ ∠3, then ∠1 is supplementary to ∠3.
peters proof:
by the linear - pair theorem, ∠1 is supplementary to ∠2. so, m∠1 + m∠2 = 180°. since ∠2 ≅ ∠3, then ∠2 = ∠3. applying the transitive property of equality, m∠1 + m∠3 = 180°, which means ∠1 is supplementary to ∠3.
vivians proof:
suppose ∠1 is not supplementary to ∠3. so, m∠1 + m∠3 ≠ 180°. by the linear - pair theorem, ∠1 is supplementary to ∠2. by the definition of supplementary angles, m∠1 + m∠2 = 180°. applying the transitive property, m∠1 + m∠3 ≠ m∠1 + m∠2. by the subtraction property of equality, this implies that m∠3 ≠ m∠2. by the definition of congruence, m∠3 ≠ m∠2. however, m∠3 ≠ m∠2 contradicts the given.
what type of proofs did they use?

Explanation:

Peter started with known facts and used logical steps to directly prove the statement, which is a direct proof. Vivian assumed the opposite of what needed to be proven and showed that it led to a contradiction with the given information, which is a proof by contradiction.

Answer:

Peter used a direct proof because he used the linear - pair theorem and properties of equality to directly show that if ∠2≅∠3, then ∠1 is supplementary to ∠3. Vivian used a proof by contradiction because she assumed ∠1 is not supplementary to ∠3 and showed that it led to a contradiction with the given congruence of ∠2 and ∠3.