Question

which is the correct reason for statement 2 in the proof?
given: ( mangle1 = mangle4 )
prove: ( mangle2 = mangle3 )

addition property
substitution property
none of the above.
linear pair theorem
given

Explanation:

Step1: Recall Linear Pair Theorem

Linear Pair Theorem states that if two angles form a linear pair, then they are supplementary (sum to \(180^{\circ}\)).

Step2: Analyze \(\angle3\) and \(\angle4\)

\(\angle3\) and \(\angle4\) form a linear pair. So, by the Linear Pair Theorem, \(m\angle3 + m\angle4=180^{\circ}\). But the first reason (for statement 1) is already given as Linear Pair Theorem. Since statement 2 is \(m\angle3 + m\angle4 = 180\), and we know from the Linear Pair Theorem (not from addition, substitution or just given as in the options) that \(m\angle3 + m\angle4=180^{\circ}\) (the first - level application of the theorem for \(\angle3\) and \(\angle4\) pair). The options provided (Addition Property: \(a=b\Rightarrow a + c=b + c\); Substitution Property: if \(a = b\) and \(b=c\), then \(a = c\); Given: already used for \(m\angle1=m\angle4\)) do not match the correct reason (Linear Pair Theorem for \(\angle3\) and \(\angle4\) pair, but the first reason is also Linear Pair Theorem for another pair. So the answer is None of the above as the options do not correctly label the reason for statement 2 considering the structure of the proof).

Answer:

None of the above.