Intersecting Angles

Date: 08/26/98 at 22:52:07
From: Todd
Subject: Geometry

Draw a diagram in which the intersection of angle AEF and angle DPC is 
ray ED. My question is how can these two angles intersect if they 
don't have any points in common? Do they intersect? Where? 

Date: 08/26/98 at 23:54:36
From: Doctor Pat
Subject: Re: Geometry

Todd, 

They do have points in common, but they didn't use them to name the 
object. Think for a moment about the line containing points P, Q, R, 
and S in order. The ray PQ will contain R and S although we do not 
need to use them to name the ray.  

In your problem it is clear that since rays EA and EF make up one 
angle, and the ray ED is the intersection with AEF and DPC, one of 
EA or EF must be the same as ED. So pick D to be along one of these.  
D is also on the ray EA. What do we know about P and C?  Well, 
since ray EA, also known as ray ED, is the intersection of both angles, 
ED must be a subset of one of the two rays that make up DPC, and since 
it contains D, it would have to be the ray PD, with point C off to the 
side somewhere.  

Now the only problem is where to put P. If P is not the same as E then 
it must be on ray ED or off of it, but if it were located somewhere on 
the ray ED, the intersection would not contain point E, so it must be 
that point E lies on the ray PD.  

Here is a picture:

       C              F
      /              /
     /              /             
    /              /
   P--------------E---------------D-------------A  
                 
I hope that helps a little. Of course it could be that C is down below 
or off to the left or someplace, but the other two rays must not 
intersect or else there would be another point in the intersection.   

Thanks for sending the question. I had to really think hard to figure 
that one out. Thanks for the challenge.

- Doctor Pat, The Math Forum
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