Short Answer 
Parallel lines are defined as lines that maintain a constant distance apart and never intersect, having equal slopes and differing y-intercepts. In geometric reasoning, when a transversal intersects two parallel lines, properties like congruent alternate exterior angles can be used to substantiate their parallelism.
Step 1: Understand Parallel Lines
Parallel lines are defined as two lines that maintain a constant distance between them and never intersect. This means they never meet, regardless of how far they are extended. The key characteristics of parallel lines are:
- They are always the same distance apart.
- They have the same slope in a coordinate system.
- They will never cross each other.
Step 2: Conditions for Parallelism
In geometry, two lines can be determined as parallel when specific conditions are met regarding their slopes and y-intercepts. The essential conditions for two lines to be parallel are:
- The slopes (m) of the two lines must be equal (m‚ÄöCA = m‚ÄöCC).
- The lines must have different y-intercepts to ensure they are not the same line.
Step 3: Applying Reasoning with Transversals
When a transversal line intersects two parallel lines, several geometric properties can be deduced. For example, properties include alternate exterior angles being congruent or vertical angles being equal. Applying these properties helps reinforce the conclusion about parallel lines:
- m || n (lines m and n are parallel).
- m‚ÄöCA ‚ÄöaO m‚ÄöCa (alternate exterior angle theorem).
- m‚ÄöCa ‚ÄöaO m‚ÄöCO (vertical angle theorem).
- m‚ÄöCA ‚ÄöaO m‚ÄöCa (transitive property). Thus, the missing statement in the proof is m‚ÄöCa ‚ÄöaO m‚ÄöCO.