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/
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