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

#1**+1 **

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

CPhill
Feb 19, 2018