Alternate and Corresponding Angles

Date: 10/21/96 at 19:20:0
From: Wesley Battersby
Subject: Geometry

Please explain corresponding and alternate angles.

Thanks, Wesley B.

Date: 02/11/97 at 21:14:48
From: Doctor Sydney
Subject: Re: Geometry

Hello Wesley,

Let's first look at a picture that we can refer to when we define 
corresponding angles and alternate angles:
 
                /
            A  /  B
    ----------/----------- 
           C /  D
            /
           /     
          /
       E / F
   -----/------------ 
     G /  H
      /

There are a lot of letters in this diagram!  Don't worry, though, 
we'll figure out what everything means in plenty of time.  

Assume that the two horizontal lines are parallel.  The diagonal is 
called a transversal, and as you can see, the intersection of the 
transversal with the horizontal lines makes lots of different angles.  
In my picture, I gave each of these angles a name:  A, B, C, D, E, F, 
G, and H. 

Whenever you have a set-up like the above in which you have two 
parallel lines and a transversal intersecting them, you can think 
about corresponding angles and alternate angles.  

Two angles are corresponding angles if they are at corresponding spots 
in the diagram. Look at the diagram. Do you see how we could easily 
split the angles into two groups?  A, B, C, and D would be the first 
group - they are the angles the transversal makes with the highest 
horizontal line.  E, F, G, and H would be the second group - they are 
the angles the transversal makes with the lowest horizontal line.  

Can you see how the bottom set of four angles looks a lot like the top 
set of four angles?  We say that two angles are corresponding angles 
if they occupy corresponding positions in the two sets of angles.  For 
example, A and E are corresponding angles because they are both in the 
"top-left" position: A is in the top left corner of the set of angles 
A, B, C, and D, while E is also in the top left corner of the set of 
angles E, F, G, and H.  

Similarly, C and G are corresponding angles.  There are two more pairs 
of corresponding angles in the picture.  Can you find them?  

One neat and helpful fact about corresponding angles is that they are 
always equal.  Can you see why? 

Let's move on to alternate angles.  Here is another copy of the 
picture I drew above:  

                /
            A  /  B
    ----------/----------- 
           C /  D
            /
           /     
          /
       E / F
   -----/------------ 
     G /  H

We say that two angles are alternate angles if they fulfill three 
requirements:

1. They must be on the "inside" or middle part of the picture.  By                  
   inside or middle angles I mean angles C, D, E, and F.
  
2. They must be on opposite sides of the transversal.  Hence A and C 
   cannot be alternate angles because they are both to the left of the 
   transversal.
  
3. If two angles are alternate, one must be from the group of angles 
   which has the top horizontal line as one of its sides and the other 
   angle must be from the group of angles which has the bottom 
   horizontal line as one of its sides.  In other words, the last 
   requirement says that a pair of alternate angles must consist of 
   one angle from the set {A, B, C, D}, and one angle from the set 
   {E, F, G, H}. 

This sounds complicated, but if we look at the picture and apply the 
three requirements, it will become clear what we mean by alternate 
angles.  

 - The first requirement tells us that only C, D, E, and F can 
   be members of a pair of angles which are parallel.  That rules 
   out a lot of possibilities.  

 - The second requirement tells us that a pair of alternate angles 
   must be on opposite sides of the transversal. So, C and E cannot 
   be a pair of alternate angles. Similarly, D and F cannot be a pair 
   of alternate angles.  

 - Applying the final constraint, we see that there are exactly two 
   pairs of alternate angles in the above picture. One pair is 
   C and F.  C and F fulfill all the requirements of alternate 
   angles: they are "inside" angles, they are on opposite sides of 
   the transversal, and they come from different groups of angles.  
   Can you find the other pair of alternate angles?

A helpful fact about alternate angles is that they, too, are equal.  
This fact can make proofs much easier!  Can you see why they are 
equal?   

I hope this helps.  Please do write back if you have more questions.

-Doctor Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/