Question

given: m || n and p is a transversal
prove: m∠2 = m∠7
what is the missing reason in the proof?
statement

1. m || n and p is a transversal
2. ∠2 ≅ ∠3
3. m∠2 = m∠3
4. ∠3 ≅ ∠7
5. m∠3 = m∠7
6. m∠2 = m∠7

reason

1. given
2. ver. ∠s theorem
3. def. of congruent
4. corr. ∠s theorem
5. def. of congruent
6. ?

corresponding angles theorem
alternate interior angles theorem
transitive property
subtraction property

Explanation:

Step1: Analyze the relationship between angles

We know that if two parallel lines \(m\parallel n\) and a transversal \(p\), then \(\angle3\cong\angle7\) (corresponding angles theorem). And \(m\angle2 = m\angle3\) (vertical angles theorem, since \(\angle2\) and \(\angle3\) are vertical angles).

Step2: Use the transitive property

The transitive property of equality states that if \(a = b\) and \(b = c\), then \(a=c\). Here, since \(m\angle2=m\angle3\) and \(m\angle3 = m\angle7\), by the transitive property, we can conclude that \(m\angle2=m\angle7\).

Answer:

transitive property