Angles of a Cyclic Quadrilateral

Date: 07/14/98 at 22:22:55
From: Bettina Yan 
Subject: Geometry of circle

ABCD is a cyclic quadrilateral with AB parallel to DC. Angle DAC is 
equal to 40 degrees, angle ABD is equal to 30 degrees. Find angles ABC, 
ADC, and ADB. Give reasons.

I have tried this question many times and am still unsure of how to do 
it and how to write out the reasons. Could you please help me?

Thanks heaps,
Bettina Yan

Date: 07/22/98 at 11:15:23
From: Doctor Rick
Subject: Re: Geometry of circle

Hi, Bettina, 

Let's get you started on this problem. The first thing, of course, is 
to draw a picture, and you can do this better than I can here:

               *  *
          A______E______B
       *  / \    |  30/ \  *
     *   / 40  \ | /     \   * 
    *   /       /|\       \   *
   *   /     /   |   \     \   *
   *  /   /      |      \   \  *
   * / /         O         \ \ *
   */____________|____________\*
   D             F             C
    *                         *
     *                       *
       *                   *
          *             *
              *  *  *

The first thing to notice is that a cyclic quadrilateral with opposite 
sides parallel is an isosceles trapezoid. I drew the perpendicular 
bisectors of AB and CD to make this clear: they are coincident, because 
they are perpendicular to parallel lines and they both pass through the 
center of the circle. (Do you understand why they pass through the 
center?) The figure is symmetrical about this line - if you need to 
prove this, you can add lines DE and CE and prove some congruent 
triangles. If you have already seen this proof, all you have to say is 
"a cyclic quadrilateral with opposite sides parallel is an isosceles 
trapezoid," and forget about line EF.

Knowing this symmetry, you can fill in the measures of two more angles
immediately - BAC and DBC. Then, by summing angles, you have DAB and 
ABC. From angle DAB you know angle ADC - why? Finally, you can find 
angle ADB by looking at triangle ADB - why?

There you have an outline of the steps you need to do. Just write them 
down in order and explain where you got them. It doesn't sound as if 
you need a formal proof, so saying, for example, "Angle BAD = angle ABD 
by symmetry" is probably enough.

If you have any more questions, feel free to ask.

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