Question

pay is working on a proof that consecutive interior angles are supplementary
two - column proof of consecutive interior angles are supplementary
statement
1 ( mangle a+mangle c = 180^{circ})
2 ( mangle a=mangle e)
3 ( mangle e+mangle c = 180^{circ})
4 ( angle e) and ( angle c) are supplementary
reason
linear pairs are supplementary
?
substitution
definition of supplementary angles
what is the missing reason in this proof?
vertical angles theorem
definition of congruence
corresponding angles postulate
transitive property of equality

Explanation:

Step1: Analyze the proof steps

In the proof, we have \(m\angle A + m\angle C=180^{\circ}\) (linear - pair supplementary). Then we need to justify \(m\angle A = m\angle E\).

Step2: Recall angle - related theorems

• Vertical Angles Theorem: Deals with angles formed by two intersecting lines (not relevant here as \(\angle A\) and \(\angle E\) are not vertical angles).
• Definition of Congruence: States that if two angles have the same measure, they are congruent. But we need a reason to show \(m\angle A = m\angle E\) first.
• Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then corresponding angles are congruent. Here, assume the two lines (the ones that \(m\) and \(n\) might be related to, based on the figure structure for the interior - angles - supplementary proof) are parallel and cut by a transversal. \(\angle A\) and \(\angle E\) are in corresponding positions.
• Transitive Property of Equality: \(a = b\) and \(b = c\) implies \(a = c\). Not relevant for getting \(m\angle A=m\angle E\) from the given proof structure.

So, the missing reason for \(m\angle A = m\angle E\) is the Corresponding Angles Postulate.

Answer:

C. Corresponding Angles Postulate