Question

1. $mangle1 = mangle5$ definition of congruent angles
2. $angle1$ and $angle2$ are a linear pair. $angle5$ and $angle8$ are a linear pair. definition of a linear pair
3. $angle1$ and $angle2$ are supplementary. $angle5$ and $angle8$ are supplementary. linear pair postulate
4. $mangle1 + mangle2 = 180^circ$ $mangle5 + mangle8 = 180^circ$ definition of supplementary angles
5. ??? ???
6. $mangle8 = 180^circ - mangle1$ $mangle2 = 180^circ - mangle1$ subtraction property of equality
7. $mangle8 = mangle2$ substitution property of equality
8. $angle8 cong angle2$ definition of congruent angles

sarah thinks the missing statement and reason should be $mangle5 + mangle2 = 180^circ$ because of the definition of supplementary angles.
lily thinks the missing statement and reason should be $mangle1 + mangle8 = 180^circ$ because of the substitution property of equality.
the student with the correct solution is .

Explanation:

Step1: Analyze Sarah's Reasoning

Sarah proposes \( m\angle5 + m\angle2 = 180^\circ \) with the definition of supplementary angles. But we know \( m\angle1 = m\angle5 \) (from step 3) and \( m\angle1 + m\angle2 = 180^\circ \) (step 6). However, the definition of supplementary angles requires a linear pair or a known supplementary relationship, and there's no direct info that \( \angle5 \) and \( \angle2 \) are supplementary, so Sarah's reason is incorrect.

Step2: Analyze Lily's Reasoning

Lily proposes \( m\angle1 + m\angle8 = 180^\circ \) with the substitution property. From step 6, we have \( m\angle5 + m\angle8 = 180^\circ \), and from step 3, \( m\angle1 = m\angle5 \). Substituting \( m\angle1 \) for \( m\angle5 \) in \( m\angle5 + m\angle8 = 180^\circ \) gives \( m\angle1 + m\angle8 = 180^\circ \), which uses the substitution property of equality. So Lily's reasoning is correct.

Answer:

Lily (the student with the correct solution is Lily, as her reasoning follows the substitution property of equality using the known equal angles and supplementary angle definitions from previous steps)