Question

given: m || n and a || b || c
prove: ∠1 ≅ ∠4
statements

1. m || n and a || b || c
2. ∠1 ≅ ∠2
3. ∠2 ≅ ∠3
4. ∠3 ≅ ∠4
5. ∠1 ≅ ∠4

reasons

1. given

2.
3.
4.
5.

Explanation:

Step1: Identify corresponding - angles

Since \(m\parallel n\) and \(a\parallel b\parallel c\), \(\angle1\) and \(\angle2\) are corresponding angles. When two parallel lines (\(m\) and \(n\)) are cut by a transversal (\(c\)), corresponding - angles are congruent. So, \(\angle1\cong\angle2\) because of the Corresponding Angles Postulate.

Step2: Identify vertical - angles

\(\angle2\) and \(\angle3\) are vertical angles. Vertical angles are always congruent. So, \(\angle2\cong\angle3\) because vertical angles are congruent.

Step3: Identify corresponding - angles

\(\angle3\) and \(\angle4\) are corresponding angles. When two parallel lines (\(a\) and \(b\)) are cut by a transversal (\(m\)), corresponding - angles are congruent. So, \(\angle3\cong\angle4\) because of the Corresponding Angles Postulate.

Step4: Use the Transitive Property

Since \(\angle1\cong\angle2\), \(\angle2\cong\angle3\), and \(\angle3\cong\angle4\), by the Transitive Property of Congruence (if \(A\cong B\) and \(B\cong C\) and \(C\cong D\), then \(A\cong D\)), we can conclude that \(\angle1\cong\angle4\).

Answer:

1. Corresponding Angles Postulate
2. Vertical angles are congruent
3. Corresponding Angles Postulate
4. Transitive Property of Congruence