peters proof: by the linear pair theorem, $ \angle 1 $ is supplementary to $ \angle 2 $. so, $ m\angle 1 + m\angle 2 = 180^{\circ} $. since $ \angle 2 \cong \angle 3 $, then $ \angle 2 = \angle 3 $. applying the transitive property of equality, $ m\angle 1 + m\angle 3 = 180^{\circ} $, which means $ \angle 1 $ is supplementary to $ \angle 3 $. vivians proof: suppose $ \angle 1 $ is not supplementary to $ \angle 3 $. so, $ m\angle 1 + m\angle 3 \
eq 180^{\circ} $. by the linear pair theorem, $ \angle 1 $ is supplementary to $ \angle 2 $. by the definition of supplementary angles, $ m\angle 1 + m\angle 2 = 180^{\circ} $. applying the transitive property, $ m\angle 1 + m\angle 3 \
eq m\angle 1 + m\angle 2 $. by the subtraction property of equality, this implies that $ m\angle 3 \
ot\cong m\angle 2 $. by definition of congruence, $ m\angle 3 \
ot\cong m\angle 2 $. however, $ m\angle 3 \cong m\angle 2 $ contradicts the given. what type of proofs did they use? peter used because. vivian used because

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# Brief Explanations:
Peter's proof starts with known theorems (linear - pair theorem) and uses substitution (since $\angle2\cong\angle3$) to directly show that $m\angle1 + m\angle3=180^{\circ}$. Vivian's proof starts by assuming the opposite of what needs to be proven ($\angle1$ is not supplementary to $\angle3$) and then shows that this assumption leads to a contradiction ($m\angle3
eq m\angle2$ which contradicts $\angle2\cong\angle3$).

# Answer:
Peter used "direct proof" because "evidence is used to support the conclusion". Vivian used "indirect proof" because "the final statement is a contradiction".

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