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### 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,

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/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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