Limiting parallel

<h2 id="definition">Definition</h2>

<p>A <a href="/facts/Ray_(geometry)/gJqsr4Gj">ray</a> 
  
    
      
        A
        a
      
    
    {\displaystyle Aa}
  
 is a limiting parallel to a ray 
  
    
      
        B
        b
      
    
    {\displaystyle Bb}
  
 if they are <a href="/facts/Coterminal_angles/gFUTEQYq">coterminal</a> or if they lie on distinct lines not equal to the line 
  
    
      
        A
        B
      
    
    {\displaystyle AB}
  
, they do not meet, and every ray in the interior of the angle 
  
    
      
        B
        A
        a
      
    
    {\displaystyle BAa}
  
 meets the ray 
  
    
      
        B
        b
      
    
    {\displaystyle Bb}
  
.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="properties">Properties</h2>
<p>Distinct lines carrying limiting parallel rays do not meet.
</p>
<h3>Proof</h3>
<p>Suppose that the lines carrying distinct parallel rays met. By definition they cannot meet on the side of 
  
    
      
        A
        B
      
    
    {\displaystyle AB}
  
 which either 
  
    
      
        a
      
    
    {\displaystyle a}
  
 is on. Then they must meet on the side of 
  
    
      
        A
        B
      
    
    {\displaystyle AB}
  
 opposite to 
  
    
      
        a
      
    
    {\displaystyle a}
  
, call this point 
  
    
      
        C
      
    
    {\displaystyle C}
  
. Thus 
  
    
      
        ∠
        C
        A
        B
        +
        ∠
        C
        B
        A
        <
        2
        
           right angles
        
        ⇒
        ∠
        a
        A
        B
        +
        ∠
        b
        B
        A
        >
        2
        
           right angles
        
      
    
    {\displaystyle \angle CAB+\angle CBA<2{\text{ right angles}}\Rightarrow \angle aAB+\angle bBA>2{\text{ right angles}}}
  
. Contradiction.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Horocycle/sBR9wnuu">horocycle</a>, In <a href="/facts/Hyperbolic_geometry/DjTMKdgy">Hyperbolic geometry</a> a <a href="/facts/Curve/xBQ0pTBW">curve</a> whose <a href="/facts/Normal_(geometry)/jhtLIhcu">normals</a> are limiting parallels</li>
<li><a href="/facts/Angle_of_parallelism/gsCvkgeh">angle of parallelism</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Hartshorne, Robin (2000). Geometry: Euclid and beyond (Corr. 2nd print. ed.). New York, NY [u.a.]: Springer. ISBN 978-0-387-98650-0. <a href="978-0-387-98650-0" target="_blank">978-0-387-98650-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Limiting parallel open-in-new

Limiting parallel