+84 1)Given angle APS = 77 degrees and arc AP = 123 degrees, find the degree measure of arc PS.
Picture: https://latex.artofproblemsolving.com/0/a/0/0a057c1f8d762cd2d19d600e3c40973008a751d4.png
2)In the figure below, EF is a diameter of the circle. Arc AE = 35 degrees and arc CF = 80 degrees. What is the measure of angle ABC, in degrees?
Picture: https://latex.artofproblemsolving.com/7/3/5/735caf3622b17bc39b7fa78bc3e915b21bf03ada.png
3)In the diagram, angle A = 30 degrees, arc DE = 170 degrees, and arc BC = 110 degrees. Find the measure of arc CE, in degrees.
Picture: https://latex.artofproblemsolving.com/0/e/6/0e62a1f5fc205df67645f931458161b433ac376c.png
4)In circle O, AP is a diameter of length 17 and PS is chord of length 8. What is the length of chord AS?
5)Grogg draws the following figure. It so happens that angle SAP is 174 degrees less than 4 times angle SOP. Find the degree measure of angle SAP.
Picture: https://latex.artofproblemsolving.com/e/f/d/efd56f87e46474e648b87ea7885f5e3471c4829a.png
6)In the diagram, minor arc AB : minor arc BC : major arc CA = 1 : 3 : 5. What is angle ABC in degrees?
Picture: https://latex.artofproblemsolving.com/7/e/d/7ed0b93f607ecc73249736f0ba6deb9edccf4df4.png
7)In the figure, if MR = MK, the measure of arc MK is 130 degrees, and measure of arc MQ is 28 degrees, then what is angle RPK, in degrees?
Picture: https://latex.artofproblemsolving.com/e/4/d/e4df975e6b7a05fe15919dc8fd88203dd3c5fa8b.png
8)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?
9)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
10)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
11)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
+128408 1)Given angle APS = 77 degrees and arc AP = 123 degrees, find the degree measure of arc PS.
If angle APS = 77°, then arc AS is twice this = 154°
So...arc PS = (360 - 123 - 154) = 83°
2)In the figure below, EF is a diameter of the circle. Arc AE = 35 degrees and arc CF = 80 degrees. What is the measure of angle ABC, in degrees?
Angle AFE = (1/2)(35°) = 17.5°
Angle CEF = (1/2)(80°) = 40°
So ....angle EBF = angle ABC = (180 - 17.5 - 40) = 122.5°
3)In the diagram, angle A = 30 degrees, arc DE = 170 degrees, and arc BC = 110 degrees. Find the measure of arc CE, in degrees.
The sum of arcs CE + DB = (360 - 110 - 170) = 80”
And angle A = (1/2) (arc CE - arc DB).....so....
30 = (1/2)(arc CE - arc DB)
60 = CE - DB
So
CE + DB = 80
CE - DB = 60 add these
2CE = 140 divide both sides by 2
arc CE = 70°
4)In circle O, AP is a diameter of length 17 and PS is chord of length 8. What is the length of chord AS?
AS will form a leg of a right triangle with AP the hypotenuse and PS the other leg....so
AS = √ [ 17^2 - 8^2 ] = √ [ 289 - 64 ] = √ [ 225] = 15 units
5)Grogg draws the following figure. It so happens that angle SAP is 174 degrees less than 4 times angle SOP. Find the degree measure of angle SAP.
SAP + 174 = 4*SOP
But SAP and SOP intercept the same arc PS...so they are equal
So...substituting
SAP + 174 = 4*SAP subtract SAP from both sides
174 = 3*SAP dvide both sides by 3
58° = SAP
6)In the diagram, minor arc AB : minor arc BC : major arc CA = 1 : 3 : 5. What is angle ABC in degrees?
There are 1 + 3 + 5 = 9 equal arcs in the circle....and angle ABC spans 5/9 of these
So major arc CA = (5/9) (360) = (360/9) * 5 = 40 * 5 = 200°
And since angle ABC is an inscribed angle intercepting this arc, it is 1/2 of this = 100°
7)In the figure, if MR = MK, the measure of arc MK is 130 degrees, and measure of arc MQ is 28 degrees, then what is angle RPK, in degrees?
Since MR = MK, then arc MR = arc MK,,,,and the sum of these arcs is 260”
So minor arc RK = 360 - 260 = 100°
So...since angle RMK intercepts this arc it has 1/2 of its measure = 50°
And since arc MQ is 28, then angle QKM = 14
So in triangle PKM angle KPM = 180 - 14 - 50 = 116°.....and angle RPK is supplemental to this = 180 - 116 = 64°
+128408 8)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?
What is the order of the arithmetic sequence ???
9)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.
AOB in the smaller circle is also 80°
So.....it's intercepted ard is twice this = 160°
