Angles in a Diagram

Date: 10/14/98 at 19:02:54
From: Elizabeth Crawford
Subject: Geometry

I was wondering how you would do a question that had perpendicular, 
parallel, and transversal lines all sort of connected together in a 
diagram and labeled with letters, and you have to find the degrees of 
the angles. They give you some of the angles' degrees. Thanks!

Date: 10/22/98 at 00:14:36
From: Doctor Santu
Subject: Re: Geometry

Hi, Elizabeth!

Isn't that nice of them to give you some of the degrees? Actually, it 
isn't as bad as it sounds. I'll give you an example. I bet you'll find 
most of this familiar from math class.

                  P
                 /\     R
                /  \   /
               /*   \ /
   A---------D/------X---------------H
             /      / \
            /      /   \
   B------C/-----E/----F\-----------G

Suppose CP and ER are parallel, and AH and BG are parallel too, and CD, 
ER and FQ are transversals to AH and BG.

Suppose they gave you the measure of the angle I have marked with a *, 
namely PDX.

You must know that "vertical angles" are equal (or "congruent," as 
they say). What is the vertical angle that goes with PDX?  When two 
lines cross each other to form sort of a letter "x", the opposite 
angles of the letter "x" are equal; in this case PDX and ADC.

You must know that "corresponding angles" are congruent. Corresponding 
angles are found when parallel lines are crossed with a transversal. 
Is PDX one of a pair of corresponding angles? Yes: PDX and DCE.

                  P
                 /\     R
                /  \   /
               /*   \ /
   A---------D/------X---------------H
            */      / \
            /*     /   \
   B------C/-----E/----F\-----------G

Now, if ER and CP are parallel, and AH and BG are parallel, then PDX 
and RXH are corresponding angles, too. So they're congruent:

                  P
                 /\     R
                /  \   /
               /*   \ /*
   A---------D/------X---------------H
            */      / \
            /*     /   \
   B------C/-----E/----F\-----------G

Now that we know RXH, we can find its vertical angles, we can find its 
corresponding angles, and so on. Each time you find an angle, check if 
it has angles congruent to it, (vertical, corresponding, alternating, 
etc., angles) and then for each of those, you find more.

Hope this helps!

- Doctor Santu, The Math Forum
  http://mathforum.org/dr.math/