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1)ABCD is a cyclic quadrilateral. Angle A, angle B, and angle C form an arithmetic sequence in this order. What is angle D in degrees?

 

2)Points A and B are on circle O such that arc AB is 80 degrees. A circle is constructed that passes through A, B, and O. Find the measure of arc AOB on this circle.

Picture: https://latex.artofproblemsolving.com/9/4/3/943110cf5221b4442f591c24df4ab3be21ca0295.png

 

3)In rectangle ABCD, we have AD = 3 and AB = 4. Let M be the midpoint of AB, and let X be the point such that MD = MX, angle MDX = 77 degrees, and A and X lie on opposite sides of DM. Find angle XCD, in degrees.

Picture: https://latex.artofproblemsolving.com/1/b/3/1b39a7de7d5064e27d8ac7d0e00bb8b0db806199.png

 

4)H is the orthocenter of acute triangle ABC and the extensions of AH, BH, and CH intersect the circumcircle of traingle ABC at A prime, B prime and C prime. We know angle AHB : angle BHC : angle CHA = 9 : 10 : 11. Find angle AprimeBprimeCprime in degrees.

Picture: https://latex.artofproblemsolving.com/6/1/1/6119f02c59ea35f11dceab068d51811cace24c1d.png

FiestyGeco  Feb 19, 2018
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2)Points A and B are on circle O such that arc AB is 80 degrees. A circle is constructed that passes through A, B, and O. Find the measure of arc AOB on this circle.

 

If arc AB  =  80°......then  angle AOB has the same measure

 

But  in the smaller circle....if angle AOB  = 80°, then minor arc AB  =  160°  because AOB is an inscribed angle in this circle, and an inscribed angle's intercepted arc has twice the measure of the angle.....So  arc AOB  =  360 - arc AB  =    360 -  160  =  200°

 

 

FeistyGeco......I tried (1)  but I could not answer it...I have not looked at  (4)  but (3)  is a lengthy problem although I'm pretty sure I know how to do it...I'll try to post it this evening if I have time

 

 

cool cool cool

CPhill  Feb 19, 2018

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