Question

given: m||n
prove: ∠1 ≅ ∠2

(diagram of two parallel lines m, n cut by a transversal, with angles 1, 3, 2 labeled)

statements reasons

1. m||n 1) given
2. ∠1 ≅ ∠3 2) blank
3. ∠3 ≅ ∠2 3) blank
4. ∠1 ≅ ∠2 4) blank

Explanation:

Step1: Identify Reason for ∠1 ≅ ∠3

Since \( m \parallel n \) and the transversal creates ∠1 and ∠3, these are corresponding angles. Corresponding angles are congruent when lines are parallel. So the reason is "Corresponding Angles Postulate".

Step2: Identify Reason for ∠3 ≅ ∠2

∠3 and ∠2 are vertical angles. Vertical angles are always congruent. So the reason is "Vertical Angles Theorem".

Step3: Identify Reason for ∠1 ≅ ∠2

If ∠1 ≅ ∠3 and ∠3 ≅ ∠2, then by the Transitive Property of Congruence, ∠1 ≅ ∠2. The Transitive Property states that if \( a \cong b \) and \( b \cong c \), then \( a \cong c \).

Answer:

1. Corresponding Angles Postulate
2. Vertical Angles Theorem
3. Transitive Property of Congruence