By definition of linear pair and algebra properties we find that m ∠ 2 ≅ m ∠ 3.
How to prove a feature of a geometric system
In this problem we must determine the existence of a linear pair consisting in a ray and two semirrays, there is a condition: m ∠ 1 + m ∠ 3 = 180º. Linear pairs are two angles such that sum of such angles equals 180º:
Step 1: Assumptions.
m ∠ 1 + m ∠ 3 = 180º
Step 2: Definition of linear pair.
m ∠ 1 + m ∠ 2 = 180º
Step 3: Transitive property.
m ∠ 1 + m ∠ 2 = m ∠ 1 + m ∠ 3
Step 4: Compatibility with addition / Commutative, associative and modultative properties / Existence of additive inverse.
m ∠ 2 = m ∠ 3
Step 5: Definition of congruence.
m ∠ 2 ≅ m ∠ 3